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Differential Equations
Laplace TransformsC. Madigan
Nova Scotia Agricultural College, Truro, NS, Canada B2N5E3 E Mail cmadigan@nsac.ca
<Text-field style="Heading 1" layout="Heading 1">Introduction</Text-field>Laplace transforms is one of the techniques for solving differential equations that is covered in an Intoductory Differential Equations course. This worksheet is a quick introduction to the Laplace transform, the Inverse Laplace and various applications for use by students in an introductory Differential Equations course.The second section of the worksheet on further applications shows how to handle solving differential equations with a periodic driving force. Included is a technique for graphically displaying the driving force together with the solution to the differential equation. As far as I could determine there is no simple way to deal directly with this situation in MAPLE. Two problems occur. First to adequately define the periodic function in question and then to determine the Laplace Inverse when it is a periodic function. periodic fcns
<Text-field style="Heading 1" layout="Heading 1">Initialization</Text-field>with(inttrans):with(DEtools):with(plots):with(student):
<Text-field style="Heading 1" layout="Heading 1">Introduction to the Laplace transform</Text-field>By the definition of the Laplace Transform, L{f(t)} = NiMtJSRpbnRHNiQqJi0lJGV4cEc2IywkKiYlInNHIiIiJSJ0R0YtISIiRi0tJSJmRzYjRi5GLS9GLjsiIiElKWluZmluaXR5Rw==, where f(t) is defined for t \342\200\26045 0.
<Text-field style="Heading 2" layout="Heading 2">Example 1. Calculate Laplace transform of f(t) = t</Text-field>f := t -> t;
F:=int(exp(-s*t)*f(t),t=0..infinity) assuming s>0;
laplace(f(t),t,s);
<Text-field style="Heading 2" layout="Heading 2">Example 2. Evaluate L{3t - 5 sin 2t}</Text-field>Solution:By linearity, L{3 t - 5 sin 2t} = 3 L{t} - 5 L{sin 2t} soRHS:=3*int(exp(-s*t)*t,t=0..infinity)-5*int(exp(-s*t)*sin(2*t),t=0..infinity);
eval(%) assuming s > 0;laplace(3*t-5*sin(2*t),t,s);
<Text-field style="Heading 2" layout="Heading 2">Example 3 Transform of a Piecewise Defined Function</Text-field>Evaluate L{f(t)} for f(t) = 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Solution:First, define and plot this function using Maple,f:=t->piecewise(t>=0 and t<3,0,t>=3,2);
plot(f(t),t=0..5,discont=true);
Using the definition of the Laplace Transform, since f(t) is piecewise continuous, the Laplace Transform is expressed as the sum of two integrals.L1 := int(0*exp(-s*t),t = 0..3);
L2 := int(2*exp(-s*t),t = 3..infinity);
L:=L1+L2;
Again this limit is 0 provided s > 0. Hence L = NiMqKCIiIyIiIi0lJGV4cEc2IywkKiYiIiRGJSUic0dGJSEiIkYlRixGLQ==laplace(f(t),t,s);
Maple cannot find the laplace transform of a piecewise defined function directly. We can redefine our function in terms of the Heaviside step function and then Maple can determine the transform for us.f1:=convert(f(t),Heaviside);
laplace(f1,t,s);
<Text-field style="Heading 3" layout="Heading 3">Example 3b. Consider the following piecewise continuous function</Text-field>f:=t->piecewise(t<=1,0,t>=1,2*t-3);
plot(f(t),t=0..5,discont=true);
We can rewrite this function is terms of the Heaviside function The unapply command is used to convert the answer into a function of the variable t.f:=unapply(factor(convert(f(t),Heaviside)),t);
Similarly below the unapply command is used to write the laplace transform as a function of sF := unapply(laplace((2*t - 3)*Heaviside(t-1),t,s),s);
laplace(f(t),t,s);
<Text-field style="Heading 2" layout="Heading 2">Example 4. Calculate <Equation executable="false" style="Heading 2" input-equation="Laplace(t^2*sin(kt))">NiMtJShMYXBsYWNlRzYjKiYpJSJ0RyIiIyIiIi0lJHNpbkc2IyUja3RHRio=</Equation></Text-field>We have the rule that NiMvLSUoTGFwbGFjZUc2IyomKSUidEclIm5HIiIiLSUiZkc2I0YpRisqJiksJEYrISIiRipGKy0tJSNAQEc2JCUiREdGKjYjLSUiRkc2IyUic0dGKw==f:=t->sin(k*t);
Laplace(t^2*f(t))=(-1)^2*diff(laplace(f(t),t,s),s,s);
simplify(%);laplace(t^2*f(t),t,s);
<Text-field style="Heading 2" layout="Heading 2">Example 5. Calculate L{kt cos kt + sin kt}</Text-field>This problem can be done three ways.First method:Note that kt cos kt + sin kt is the derivative of t sin kt.Hence, L{kt cos kt + sin kt} = L{d/dt (t sin kt)} = sL{t sin kt} = s(- d/ds L{sin kt}) Second method: L{ d/dt(f(t)) } = s L{ f(t) } - f(0) So define f(t) and apply this formula.Third method : Use the laplace command directly.
<Text-field style="Heading 2" layout="Heading 2">Example 6. Dirac Delta function.</Text-field>The Dirac delta function is found in Maple in the form Dirac[t - LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn] where the impulse is at t =LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn. The Laplace transform package can work with it.f:='f':t:='t':Dirac(t-t[0]);
laplace(Dirac(t-2),t,s);
laplace(f(t)*Dirac(t-2),t,s);
invlaplace(exp(-3*s),s,t);
invlaplace(1/(s^2-5)*exp(-3*s),s,t);
simplify(convert(%,exp));
<Text-field style="Heading 2" layout="Heading 2">Exercises Use the various methods to calculate the Laplace transforms and use the command laplace to check your answers.</Text-field>1. Use the definition to find the Laplace transform of f(t) = -1 if 0 <= t < 1,1 if t >= 1.2. Use the definition to find the Laplace transform of f(t) = 2t + 1 if 0 <= t < 1,0 if t >= 1.3. Use the definition to find the Laplace transform of NiMvLSUiZkc2IyUidEctJSRleHBHNiMsJkYnIiIiIiIoISIi4. Use the definition to find the Laplace transform of f(t) = NiMqJiklInRHIiIjIiIiLSUkZXhwRzYjKiYiIiRGJ0YlRidGJw== .5. Find the Laplace transform of f(t) = NiMqJiIiIyIiIiokKSUidEciIiVGJUYl.6. Find the Laplace transform of f(t) = 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 = 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7. Find the Laplace transform of f(t) = cos 5t + sin 2t.8. Evaluate (a) L{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}; (c) L{t^2}sin(kt)} , and (d) L{t 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( t )}9. Find the Laplace transform of t 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.,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 cos(4t), t(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 + 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JSMlP0c= .10. Find the Laplace transform of (3t + 1)U(t - 3), (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(t - 1).11. Find the Laplace transform of (t 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)U(t - 5).12. a) Write f(t) = 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 in terms of unit step functions. Find the Laplace transform of the given function. b) Write f(t) = 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 in terms of unit step functions. Find the Laplace transform of the given function.
<Text-field style="Heading 1" layout="Heading 1">Inverse Transform</Text-field>
<Text-field style="Heading 2" layout="Heading 2">Example 1. Termwise Division and Linearity</Text-field>Evaluate L^(-1){(3 s + 5)/(s^2 + 7)}.Solution:The given function of s can be written as two expressions by means of termwise division:F := (3*s + 5)/(s^2 + 7);
F := expand(F);
LI := 3*cos(sqrt(7)*t) + 5/sqrt(7)*sin(sqrt(7)*t);
This can be done directly using the invlaplace which recognizes the linearity of the inverse Laplace transform.ls := invlaplace((3*s + 5)/(s^2 + 7),s,t);
rs := 3*invlaplace(s/(s^2 + 7),s,t) + 5/sqrt(7)*invlaplace(sqrt(7)/(s^2 + 7),s,t);
ls = rs;
<Text-field style="Heading 2" layout="Heading 2">Example 2 Partial Fractions and Linearity</Text-field>Evaluate NiMqJiklIkxHLCQiIiIhIiJGJzwjKiYsJiUic0dGJ0YnRidGJyomKUYsIiIjRicpLCZGLEYnRi9GJyIiJEYnRihGJw==Solution:LetF := s -> (s+1)/(s^2*(s+2)^3);
F1:=convert(F(s),parfrac,s);
T1:=invlaplace(op(1,F1),s,t);
T2:=invlaplace(op(2,F1),s,t);T3:=invlaplace(op(3,F1),s,t);T4:=invlaplace(op(4,F1),s,t);
y:=T1+T2+T3+T4;
Maple can also calculate the inverse of F(s) directlyinvlaplace(F(s),s,t);
<Text-field style="Heading 2" layout="Heading 2">Example 3 Completing the Square to find L^(-1)</Text-field>Evaluate L^(-1){s/(s^2 + 6 s + 11)}.Solution:Let F := (s) -> s/(s^2 + 6*s + 11);
F:=unapply(completesquare(F(s)),s);
F:=s->(s+3)/((s+3)^2+2)-3/((s+3)^2+2);
Applying the !st translation theorem we haveInv(F(s)):=exp(-3*t)*Inv(s/(s^2+2))-3/sqrt(2)*exp(-3*t)*Inv(sqrt(2)/(s^2+2));
f:=t->exp(-3*t)*cos(sqrt(2)*t)-3/2*2^(1/2)*exp(-3*t)*sin(sqrt(2)*t);
f := unapply(invlaplace(F(s),s,t),t);
<Text-field style="Heading 2" layout="Heading 2">Example 4 Convolutions</Text-field>
<Text-field style="Heading 3" layout="Heading 3">Definition</Text-field>Given two functions f(t) and g(t) their convolution f * g is defined as f*g= NiMtJSRpbnRHNiQqJi0lImZHNiMlJHRhdUciIiItJSJnRzYjLCYlInRHRitGKiEiIkYrL0YqOyIiIUYwThe Laplace{ f*g } = L{ f(t) }L{ g(t) } = F(s) G(s)
<Text-field style="Heading 4" layout="Heading 4">example</Text-field>Given f(t) =exp(t) g(t) = sin(t)f:=t->exp(t);g:=t->sin(t);
fstarg:=int(f(tau)*g(t-tau),tau=0..t);
gstarf:=int(g(tau)*f(t-tau),tau=0..t);
laplace(fstarg,t,s)=laplace(f(t),t,s)*laplace(g(t),t,s);
simplify(lhs(%));
<Text-field style="Heading 3" layout="Heading 3">Application</Text-field>Since L { f( * g } = F(s) G(s) then NiMpJSJMRywkIiIiISIi(F(s)G(s)) = f*g
<Text-field style="Heading 4" layout="Heading 4">Example Inverse Transform as a Convolution</Text-field>Evaluate L^(-1){1/((s - 1)(s + 4))}F:=s->1/(s-1); G:=s->1/(s+4);
f := unapply(invlaplace(1/(s-1),s,t),t);
g := unapply(invlaplace(1/(s+4),s,t),t);
IL := int(f(tau)*g(t - tau), tau = 0..t);
invlaplace(F(s)*G(s),s,t);
<Text-field style="Heading 2" layout="Heading 2">Exercises Using various methods calculate the Laplace inverse. Use the invlaplace command to check your answer.</Text-field>2. Find the inverse transform of 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.3. Find the inverse transform of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiMtSSZtZnJhY0dGJDYoLUkjbW5HRiQ2JFEiNUYnL0YzUSdub3JtYWxGJy1GIzYlLUklbXN1cEdGJDYlLUYsNiVRInNGJ0YvRjItRiM2Iy1GOzYkUSIyRidGPi8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYwUSIrRidGPi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGVi8lKXN0cmV0Y2h5R0ZWLyUqc3ltbWV0cmljR0ZWLyUobGFyZ2VvcEdGVi8lLm1vdmFibGVsaW1pdHNHRlYvJSdhY2NlbnRHRlYvJSVmb3JtR1EmaW5maXhGJy8lJ2xzcGFjZUdRMG1lZGl1bW1hdGhzcGFjZUYnLyUncnNwYWNlR0Ziby8lKG1pbnNpemVHUSIxRicvJShtYXhzaXplR1EpaW5maW5pdHlGJy1GOzYkUSM0OUYnRj4vJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmNwLyUpYmV2ZWxsZWRHRlZGKw==.4. Find the inverse transform of 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.5. Find the inverse transform of 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 .6. Find the inverse transform of 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.7. Find the inverse transform of 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 .8. Find the inverse transform of 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. 9. Find the inverse transform of 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.10. Find the inverse transform of 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.11. Find the inverse transform of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiMtSSZtZnJhY0dGJDYoLUkjbW5HRiQ2JFEiMUYnL0YzUSdub3JtYWxGJy1GIzYkLUklbXN1cEdGJDYlLUYsNiVRInNGJ0YvRjItRiM2Iy1GOzYkUSIzRidGPi8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRictSShtZmVuY2VkR0YkNiQtRiM2JUZFLUkjbW9HRiQ2MFEiK0YnRj4vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRmVuLyUpc3RyZXRjaHlHRmVuLyUqc3ltbWV0cmljR0Zlbi8lKGxhcmdlb3BHRmVuLyUubW92YWJsZWxpbWl0c0dGZW4vJSdhY2NlbnRHRmVuLyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUTBtZWRpdW1tYXRoc3BhY2VGJy8lJ3JzcGFjZUdGZ28vJShtaW5zaXplR0Y9LyUobWF4c2l6ZUdRKWluZmluaXR5RidGOkY+LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZkcC8lKWJldmVsbGVkR0ZlbkYr.
<Text-field style="Heading 1" layout="Heading 1">Applications</Text-field>
<Text-field style="Heading 2" layout="Heading 2">Example 1 An Initial-Value Problem</Text-field>s := 's' : t := 't' : y := 'y' :
deq := diff(y(t),t,t) - 6*diff(y(t),t) + 9*y(t) = t^2*exp(3*t);
eqn := laplace(deq,t,s);
eqn := subs( [y(0) = 2, D(y)(0) = 6], eqn);
Y :=unapply(convert(solve(eqn, laplace(y(t), t, s)),parfrac,s),s);
y := unapply(invlaplace(Y(s),s,t),t);
y := 'y' : dsolve({deq, y(0) =2, D(y)(0) = 6}, y(t), method=laplace);
<Text-field style="Heading 2" layout="Heading 2">Example 2. Using a Unit Step Function</Text-field>Solve x'' + 16 x = f(t) , x(0) = 0, x'(0) = 1 wheref(t) = 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The function f(t) can be interpreted as an external force that is acting on a mechanical system for only a short period of time and then is removed. Although this problem can be solved conventionally, it is not at all convenient when f(t) is defined in a piecewise manner.f := t -> cos(4*t) - cos(4*(t-Pi))*Heaviside(t - Pi);
plot(f(t), t = 0..2*Pi, color = red);
x := 'x' :
deq := diff(x(t),t,t) + 16*x(t) = f(t);
eqn := laplace(deq,t,s) ;
eqn := subs([x(0) = 0, D(x)(0) = 1], eqn);
X := unapply(expand(solve(eqn, laplace(x(t),t,s))),s);
x := unapply(invlaplace(X(s),s,t),s);
plot(x(t), t = 0..3*Pi, color = blue);
<Text-field style="Heading 2" layout="Heading 2">Example 3. Solving an integral equation</Text-field>Solve NiMvLSUiZkc2IyUidEcsKComIiIkIiIiKiQpRiciIiNGK0YrRistJSRleHBHNiMsJEYnISIiRjMtJSRJbnRHNiQqJi1GJTYjJSR0YXVHRistRjA2IywkRjpGM0YrRidGMw== for f(t).f := 'f' :
eq1 := f(t) = 3*t^2 - exp(-t) - int(f(tau)*exp(t-tau), tau = 0..t);
eq := laplace(eq1,t,s);
F := unapply(convert(simplify(solve(eq,laplace(f(t),t,s))),parfrac,s),s);
f:= unapply(invlaplace(F(s),s,t),t);
<Text-field style="Heading 2" layout="Heading 2">Example 4. Using the Dirac Delta function</Text-field>Solve y'' + y = 4delta(t - 2 Pi), y(0) = 1, y'(0) = 0.y:='y':deq := diff(y(t),t,t) + y(t) = 4*Dirac(t - 2*Pi);
eqn := laplace(deq,t,s);
eqn := subs( [y(0) = 1, D(y)(0) = 0] , eqn);
Y := unapply(simplify(solve(eqn, laplace(y(t), t, s))),s);
y := unapply(invlaplace(Y(s),s,t),t);
plot(y(t), t = 0..4*Pi , color = red);
y := 'y' :
dsolve({deq, y(0) = 1, D(y)(0) = 0}, y(t),method= laplace);
<Text-field style="Heading 2" layout="Heading 2">Example 5. Solving a system of equations</Text-field>Solve LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0krbXNlbWFudGljc0dGJFEnYXRvbWljRic='' + 10LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0krbXNlbWFudGljc0dGJFEnYXRvbWljRic= - 4LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0krbXNlbWFudGljc0dGJFEnYXRvbWljRic= = 0 -4LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0krbXNlbWFudGljc0dGJFEnYXRvbWljRic= + LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0krbXNlbWFudGljc0dGJFEnYXRvbWljRic='' + 4LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0krbXNlbWFudGljc0dGJFEnYXRvbWljRic= = 0subject to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0krbXNlbWFudGljc0dGJFEnYXRvbWljRic=(0) = 0, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0krbXNlbWFudGljc0dGJFEnYXRvbWljRic='(0) = 1, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0krbXNlbWFudGljc0dGJFEnYXRvbWljRic=(0) = 0, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0krbXNlbWFudGljc0dGJFEnYXRvbWljRic='(0) = -1.deq1 := diff(x1(t),t,t) + 10*x1(t) - 4*x2(t) = 0;
deq2 := -4*x1(t) + diff(x2(t),t,t) + 4*x2(t) = 0;
eqn1 := laplace(deq1,t,s);
eqn2 := laplace(deq2,t,s);
eqn1 := subs([x1(0) = 0, D(x1)(0) = 1, x2(0) = 0, D(x2)(0) = -1],eqn1);
eqn2 := subs([x1(0) = 0, D(x1)(0) = 1, x2(0) = 0, D(x2)(0) = -1],eqn2);
soln := solve({eqn1,eqn2}, {laplace(x1(t), t, s) ,laplace(x2(t), t, s)});
X1 := unapply(rhs(soln[2]),s);
X2 := unapply(rhs(soln[1]),s);
x1 := unapply(invlaplace(X1(s),s,t),t);
x2 := unapply(invlaplace(X2(s),s,t),t);
x1 := 'x1' : x2 := 'x2' :
dsolve({deq1,deq2,x1(0) = 0, D(x1)(0) = 1, x2(0) = 0, D(x2)(0) = -1},{x1(t),x2(t)});
(a)^(b);
<Text-field style="Heading 2" layout="Heading 2">Exercises</Text-field>Solve the following IVPs1. y' + 4y = 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 , y(0) = 2.2. y'' - 6y' + 13y = 0, y(0) = 0, y'(0) = -3.3. y'' - 4y' + 4y = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiM0YnL0Y2USdub3JtYWxGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRic=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, y(0) = 0, y'(0) = 0.4. y'' + y = sin t, y(0) = 1, y'(0) = -1.5. y' + y = f(t) where f(t) = 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 , y(0) = 0.6. y'' + 4y = sin( t ) U(t - 2x0), y(0) = 1, y'(0) =0.7. NiMsJi0lImZHNiMlInRHIiIiKiYiIiNGKC0lJEludEc2JComLSUkY29zRzYjLCZGJ0YoJSR0YXVHISIiRigtRiU2I0YzRigvRjM7IiIhRidGKEYo = NiMsJiomIiIlIiIiLSUkZXhwRzYjLCQlInRHISIiRiZGJi0lJHNpbkc2I0YrRiY=8. y'' + 16 y = delta(t - 2 x0), y(0) = 0, y'(0) = 0.9. y'' + y = delta(t - LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiMtSSZtZnJhY0dGJDYoLUYsNiVRJSZwaTtGJy9GMFEmZmFsc2VGJy9GM1Enbm9ybWFsRictRiM2Iy1JI21uR0YkNiRRIjJGJ0Y/LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZMLyUpYmV2ZWxsZWRHRj5GKw==) + delta(t - 3LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiMtSSZtZnJhY0dGJDYoLUYsNiVRJSZwaTtGJy9GMFEmZmFsc2VGJy9GM1Enbm9ybWFsRictRiM2Iy1JI21uR0YkNiRRIjJGJ0Y/LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZMLyUpYmV2ZWxsZWRHRj5GKw==), y(0) = 0, y'(0) = 0.10. dx/dt = -x + y, dy/dt = 2x, x(0) = 0, y(0) = 1.11. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbWZyYWNHRiQ2KC1GIzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiMkYnL0Y7USdub3JtYWxGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRictRjQ2JVEieEYnRjdGOi1GIzYjLUYxNiUtRjQ2JVEjZHRGJ0Y3RjpGPUZFLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZXLyUpYmV2ZWxsZWRHUSZmYWxzZUYnLUY0NiVRIUYnRjdGOg== + x - y = 0, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkmbWZyYWNHRiQ2KC1GIzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiMkYnL0Y7USdub3JtYWxGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRictRjQ2JVEieUYnRjdGOi1GIzYjLUYxNiUtRjQ2JVEjZHRGJ0Y3RjpGPUZFLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZXLyUpYmV2ZWxsZWRHUSZmYWxzZUYn + y - x = 0, x(0) = 0, x'(0) = -2, y(0) = 1, y'(0) = 1.12. dx/dt - 4 x + 2 y = 2 U(t-1), dy/dt + y - 3x = U(t-1), x(0) = 0, y(0) = 1/2.
<Text-field bookmark="applic periodic fcn" style="Heading 1" layout="Heading 1">Further applications</Text-field>
<Text-field style="Heading 2" layout="Heading 2">Introduction</Text-field>This section contains a set of solutions to differential equations directly and using Maples dsolve.It also contains a method to define a periodic function using a piecewise function definition so that one can obtain a visualization of the periodic function and then using this structure solve a differential equation having a periodic driving force.
<Text-field style="Heading 2" layout="Heading 2">Initialization</Text-field>with(inttrans):with(DEtools):with(plots):with(student):
<Text-field style="Heading 2" layout="Heading 2">Applications</Text-field>If you need assistance with Laplace transforms click here Intro to LPT
<Text-field style="Heading 3" layout="Heading 3">Example 1 Solving an IVP.</Text-field>Solve NiMvLCgtLS0lI0BARzYkJSJERyIiIzYjJSJ5RzYjJSJ0RyIiIiomIiIlRjAtLUYqRixGLkYwRjAqJiIiJ0YwLUYtRi5GMEYwLCZGMEYwLSUkZXhwRzYjLCRGLyEiIkYw, y(0)=D(y)(0)=0Define the eqn and take the laplace transform of it.y:='y':t:='t':Y:='Y':eqn:=(D@@2)(y)(t)+4*D(y)(t)+6*y(t)=1+exp(-t);eqn1:=laplace(eqn,t,s);Substituting in the initial values and solving for L{y(t)}eqn2:=subs({y(0)=0,D(y)(0)=0,laplace(y(t),t,s)=Y(s)},eqn1);solve(eqn2,Y(s));invlaplace(%,s,t);Solving the eqn directly using dsolve with the option laplacedsolve({eqn,y(0)=0,D(y)(0)=0},y(t),method=laplace);Solving the eqn with dsolve but without the laplace option.dsolve({eqn,y(0)=0,D(y)(0)=0},y(t));simplify(%);
<Text-field style="Heading 3" layout="Heading 3">Example 2. Working with a piecewise continuous or periodic function</Text-field>
<Text-field bookmark="piecewise fcn" style="Heading 4" layout="Heading 4">Defining a piecewise continuous function</Text-field>Defining a piecewise 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f:=t->piecewise(t>=0 and t<2,t,t>2 and t<=4,4-t,0);plot(f(t),t=0..6);
<Text-field style="Heading 4" layout="Heading 4">Defining a periodic function</Text-field>f:=t->piecewise(t>=0 and t<2,t,t>=2 and t<4,4-t,0);f(t);Maple does not have a mechanism to handle periodic functions in general.We can make a piecewise continuous function periodic by defining another function g(k) = f(t-4k) which has a period of 4. and g(k) represents the kth period.Making f (t) defined in the previous section into a periodic function.g:=k->f(t-4*k);g(1);Plotting the first 4 periods of our periodic function.plot({g(k) $k=0..3},t=0..20);
<Text-field style="Heading 4" layout="Heading 4">Calculating a Laplace transform for a piecewise or periodic function</Text-field>In order to calculate the laplace transform we first convert f(t) and g(t) using the Heaviside functionconvert(f(t),Heaviside);laplace(%,t,s);invlaplace(%,s,t);convert(%,piecewise);convert(g(k),Heaviside);gl:=laplace(%,t,s);glv:=invlaplace(gl,s,t);convert(glv,piecewise,t);Plotting the first 11 periods of g(k)plot({glv $k=0..10},t=0..50);
<Text-field style="Heading 4" layout="Heading 4">Solving an IVP with a piecewise continuous driving function</Text-field>
<Text-field style="_pstyle18" selection-placeholder="false" executable="false" family="Times New Roman" opaque="false" foreground="[0,0,0]" superscript="false" placeholder="false" readonly="false" subscript="false" bold="false" italic="false" underline="false" background="[255,255,255]" font_style_name="_pstyle18" size="12" layout="_pstyle18" linebreak="space" spaceabove="0" rightmargin="0" bullet="none" firstindent="0" layout_style_name="_pstyle18" linespacing="0.0" pagebreak-before="false" leftmargin="0" alignment="left" initial="0" spacebelow="0"><Font selection-placeholder="false" executable="false" family="Times New Roman" opaque="false" foreground="[0,0,0]" superscript="false" placeholder="false" readonly="false" subscript="false" bold="false" italic="false" underline="false" background="[255,255,255]" style="_cstyle17" font_style_name="_cstyle17" size="12">Example 1. Solve </Font><Equation executable="false" style="_pstyle18" input-equation="(diff(y(t), t, t))+4*(diff(y(t), t))+6*y(t) = f(t)">NiMvLCgtJSVkaWZmRzYlLSUieUc2IyUidEdGK0YrIiIiKiYiIiVGLC1GJjYkRihGK0YsRiwqJiIiJ0YsRihGLEYsLSUiZkdGKg==</Equation><Font selection-placeholder="false" executable="false" family="Times New Roman" opaque="false" foreground="[0,0,0]" superscript="false" placeholder="false" readonly="false" subscript="false" bold="false" italic="false" underline="false" background="[255,255,255]" style="_cstyle17" font_style_name="_cstyle17" size="12"> where and y(0) = y '(0) = 0</Font></Text-field>Defining the DEQ using the fact that f(t) was defined in the previous section.eqn1:=(D@@2)(y)(t)+4*D(y)(t)+6*y(t)=convert(f(t),Heaviside);Applying the Laplace transformeqnlp:=laplace(eqn1,t,s);Substituting eqnlp1:=subs({y(0)=0,D(y)(0)=0,laplace(y(t),t,s)=Y(s)},eqnlp);solve(eqnlp1,Y(s));sol:=invlaplace(%,s,t);solp:=convert(sol,piecewise);plot({sol},t=0..20);
<Text-field style="_pstyle18" selection-placeholder="false" executable="false" family="Times New Roman" opaque="false" foreground="[0,0,0]" superscript="false" placeholder="false" readonly="false" subscript="false" bold="false" italic="false" underline="false" background="[255,255,255]" font_style_name="_pstyle18" size="12" layout="_pstyle18" linebreak="space" spaceabove="0" rightmargin="0" bullet="none" firstindent="0" layout_style_name="_pstyle18" linespacing="0.0" pagebreak-before="false" leftmargin="0" alignment="left" initial="0" spacebelow="0"><Font selection-placeholder="false" executable="false" family="Times New Roman" opaque="false" foreground="[0,0,0]" superscript="false" placeholder="false" readonly="false" subscript="false" bold="false" italic="false" underline="false" background="[255,255,255]" style="_cstyle17" font_style_name="_cstyle17" size="12">Example 2. Solve IVP </Font><Equation executable="false" style="_pstyle18" input-equation="(diff(y(t), t, t))+3*(diff(y(t), t))+y(t) = g(t)">NiMvLCgtJSVkaWZmRzYlLSUieUc2IyUidEdGK0YrIiIiKiYiIiRGLC1GJjYkRihGK0YsRixGKEYsLSUiZ0dGKg==</Equation><Font selection-placeholder="false" executable="false" family="Times New Roman" opaque="false" foreground="[0,0,0]" superscript="false" placeholder="false" readonly="false" subscript="false" bold="false" italic="false" underline="false" background="[255,255,255]" style="_cstyle17" font_style_name="_cstyle17" size="12"> where and y(0) = y '(0)= 1.</Font></Text-field>Defining g(t):gg:=t->piecewise(t<0,0,t>=0 and t<1,1,t>=1 and t<2,-1,t>2,0);Defining the IVPeqn:=(D@@2)(y)(t)+3*D(y)(t)+y(t)=gg(t);eqn1:=simplify(convert(eqn,Heaviside));Applying the Laplace transformleqn:=laplace(eqn1,t,s);Substituting the initial values and solving for Y(s)Y(s):=solve(subs({y(0)=1,D(y)(0)=1},leqn),laplace(y(t),t,s));Applying the inverse Laplace transformsol:=invlaplace(Y(s),s,t);Plotting the input and solution curves together.plot({gg(t),sol},t=0..5);convert(sol,exp);convert(%,piecewise);simplify(%);
<Text-field style="_pstyle18" selection-placeholder="false" executable="false" family="Times New Roman" opaque="false" foreground="[0,0,0]" superscript="false" placeholder="false" readonly="false" subscript="false" bold="false" italic="false" underline="false" background="[255,255,255]" font_style_name="_pstyle18" size="12" layout="_pstyle18" linebreak="space" spaceabove="0" rightmargin="0" bullet="none" firstindent="0" layout_style_name="_pstyle18" linespacing="0.0" pagebreak-before="false" leftmargin="0" alignment="left" initial="0" spacebelow="0"><Font selection-placeholder="false" executable="false" family="Times New Roman" opaque="false" foreground="[0,0,0]" superscript="false" placeholder="false" readonly="false" subscript="false" bold="false" italic="false" underline="false" background="[255,255,255]" style="_cstyle17" font_style_name="_cstyle17" size="12">Example 3. Solve IVP </Font><Equation executable="false" style="_pstyle18" input-equation="(diff(y(t), t, t))+3*(diff(y(t), t))+4*y(t) = f(t)">NiMvLCgtJSVkaWZmRzYlLSUieUc2IyUidEdGK0YrIiIiKiYiIiRGLC1GJjYkRihGK0YsRiwqJiIiJUYsRihGLEYsLSUiZkdGKg==</Equation><Font selection-placeholder="false" executable="false" family="Times New Roman" opaque="false" foreground="[0,0,0]" superscript="false" placeholder="false" readonly="false" subscript="false" bold="false" italic="false" underline="false" background="[255,255,255]" style="_cstyle17" font_style_name="_cstyle17" size="12"> where y(o) = y '(0) = 0</Font></Text-field>f:=t->piecewise(t<1,0,t>=1 and t<2,cos(t),t>=2,sin(t));eqn:=(D@@2)(y)(t)+3*D(y)(t)+4*y(t)=f(t);eqn1:=simplify(convert(eqn,Heaviside));leqn:=laplace(eqn1,t,s);Y(s):=solve(subs({y(0)=0,D(y)(0)=0},leqn),laplace(y(t),t,s));sol:=invlaplace(Y(s),s,t);plot(sol,t=0..10);soln1:=dsolve({eqn1,y(0)=0,D(y)(0)=0},y(t));plot(rhs(soln1),t=0..20);
<Text-field style="Heading 4" layout="Heading 4">Solving an IVP with a periodic driving force.</Text-field>Solve NiMvLCgtJSVkaWZmRzYlLSUieUc2IyUidEdGK0YrIiIiKiYiIiVGLC1GJjYkRihGK0YsRiwqJiIiJ0YsRihGLEYsLSUiZ0dGKg== where g(t) is the periodic function defined by cycling the piecewise continuous function f(t). andg(k) = f( t - 4k ) . g has a period of 4 units JSMlP0c=f:=t->piecewise(t>=0 and t<2,t,t>2 and t<=4,4-t,0);Making f (t) into a periodic function.g:=k->f(t-4*k);defining the DEQ and using only 2 cycles of the input functioneqn2:=(D@@2)(y)(t)+4*D(y)(t)+6*y(t)=simplify(sum(convert(g(k),Heaviside),k=0..1));Applying the Laplace transformleqn2:=laplace(eqn2,t,s);Subs the initial values and solving for Y(s)Y(s):=solve(subs({y(0)=0,D(y)(0)=0},leqn2),laplace(y(t),t,s));Applying the inverse transform to obtain the solution.sol2:=invlaplace(Y(s),s,t);Plotting the input function and the solution curve on the same graph.plot({sol2,g(k) $k=0..1},t=0..10);convert(sol2,piecewise);plot(%, t = 0 .. 10);
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