ORDINARY DIFFERENTIAL EQUATIONS POWERTOOLLesson 4 -- First-Order Linear EquationsProf. Douglas B. MeadeIndustrial Mathematics InstituteDepartment of MathematicsUniversity of South CarolinaColumbia, SC 29208 URL: http://www.math.sc.edu/~meade/E-mail: meade@math.sc.eduCopyright \251 2001 by Douglas B. MeadeAll rights reserved-------------------------------------------------------------------

4.A Structure of Solutiosn to Linear ODEs4.B Integrating Factor for a First-Order Linear ODE

restart;with( DEtools ):with( plots ):

The general first-order linear ODE islin_ode := diff( x(t), t ) + p(t) * x(t) = f(t);The dependent function NiMtJSJ4RzYjJSJ0Rw== and its first derivative, NiMqJiUjZHhHIiIiJSNkdEchIiI=, appear linearly, that is, appear as a linear combination with variable coefficients. Note that, in general, this is not a separable ODE, as Maple shows viaodeadvisor( lin_ode, [separable] );If NiMtJSJwRzYjJSJ0Rw== and NiMtJSJmRzYjJSJ0Rw== are both constants, then the ODE is separable, in which case the solution can be found as in Lesson 3. The constant-coefficient linear equationlin_ode_const := subs( p(t)=a, f(t)=b, lin_ode );is verified separable by Maple viaodeadvisor( lin_ode_const, [separable] );Maple's solution process can be seen withinfolevel[dsolve] := 3:lin_ode_const_soln := dsolve( lin_ode_const, x(t) );infolevel[dsolve] := 0:The structure of this solution is important. The term involving the constant of integration, that is, the termsoln_h := x(t) = coeff(rhs(lin_ode_const_soln),_C1);is a solution of the homogenous ODE (i.e., NiMvJSJiRyIiIQ==). This is verified viaodetest( soln_h, subs( b=0, lin_ode_const ) );The constant term, namely,soln_p := x(t) = subs( _C1=0, rhs(lin_ode_const_soln) );is a solution to the non-homogeneous ODE, as is seen viaodetest( soln_p, lin_ode_const );Any solution that satisfies the full ODE is called a particular solution. It is a general property of linear equations that the general solution can be written as the sum of the general solution to the homogeneous equation and any (particular) solution to the non-homogeneous equation. This structure will appear again when linear equations of higher-order, and linear systems of first-order ODEs are studied.

Knowledge of the structure of solutions to linear ODEs is important, but does not provide too much information about finding solutions to the general first-order linear ODENiMvLCYqJiUjZHhHIiIiJSNkdEchIiJGJyomLSUicEc2IyUidEdGJy0lInhHRi1GJ0YnLSUiZkdGLQ== A procedure for solving the first-order linear ODE consists of finding an integrating factor, NiMtJSNtdUc2IyUidEc=, for the ODE. An integrating factor is a function NiMtJSNtdUc2IyUidEc= that, upon multiplication against the left-hand side of the equation, renders the product an exact derivative. Thus, the integrating factor NiMtJSNtdUc2IyUidEc= is characterized by the property thatNiMvKiYsJi0lI3gnRzYjJSJ0RyIiIiomLSUicEdGKEYqLSUieEdGKEYqRipGKi0lI211R0YoRioqJiUiZEdGKiUjZHRHISIi NiMtJSFHNiMqJi0lI211RzYjJSJ0RyIiIi0lInhHRilGKw== In other words, NiMtJSNtdUc2IyUidEc= is a factor that allows the left-hand side of the ODE to be written as the derivative of the product NiMqJi0lI211RzYjJSJ0RyIiIi0lInhHRiZGKA==. The general formula for the integrating factor for the first-order linear ODE is NiMvLSUjbXVHNiMlInRHLSUkZXhwRzYjLSUkaW50RzYkLSUicEdGJkYn.Multiplication of the ODE by this factor leads to an equation of the formNiMvLSUlZGlmZkc2JC0lIlhHNiMlInRHRiotJSJGR0Yp where NiMvLSUiWEc2IyUidEcqJi0lI211R0YmIiIiLSUieEdGJkYr and NiMvLSUiRkc2IyUidEcqJi0lI211R0YmIiIiLSUiZkdGJkYr. The explicit general solution of this equation can be found by direct integration to beNiMtJSJYRzYjJSJ0Rw== = NiMtJSRJbnRHNiQtJSVkaWZmRzYkLSUiWEc2IyUidEdGLEYs = NiMsJi0lJEludEc2JC0lIkZHNiMlInRHRioiIiIlIkNHRis=.Thus, NiMvKiYtJSJ4RzYjJSJ0RyIiIi0lI211R0YnRiksJi0lJGludEc2JComLSUiZkdGJ0YpRipGKUYoRiklIkNHRik= and the solution to the original ODE is found using NiMvLSUieEc2IyUidEcqJi0lIlhHRiYiIiItJSNtdUdGJiEiIg== = NiMqJiwmLSUkaW50RzYkKiYtJSJmRzYjJSJ0RyIiIi0lI211R0YrRi1GLEYtJSJDR0YtRi1GLiEiIg== = NiMsJiomLSUkZXhwRzYjLCQtJSRpbnRHNiQlInBHJSJ0RyEiIiIiIi1GKjYkKiYtJSJmRzYjRi1GLy1GJjYjRilGL0YtRi9GLyomJSJDR0YvRiVGL0YvInstead of focusing on the general formula, implement the solution process for each specific problem.For example, consider the specific first-order linear ODElin_ode1 := diff( x(t), t ) + x(t)/(t+1) = ln(t)/(t+1);In this problem, NiMvLSUicEc2IyUidEcqJiIiIkYpLCZGJ0YpRilGKSEiIg== and NiMvLSUiZkc2IyUidEcqJi0lI2xuR0YmIiIiLCZGJ0YrRitGKyEiIg==. Thus, the integrating factor isint_fact := mu(t) = exp( Int( 1/(t+1), t ) );which evaluates toint_fact1 := value( int_fact );The DEtools package contains intfactor, a procedure that will find an integrating factor for problems of this type. It givesintfactor( lin_ode1 );which is exactly what was obtained above.Multiplication of the ODE by this integrating factor yieldsode2 := subs( int_fact1, mu(t)*lin_ode1 );While this equation is rather complicated, the definition of the integrating factor allows us to replace the left-hand side with the single derivativeode3 := subs( int_fact1, Diff( mu(t)*x(t), t ) ) = rhs(ode2);Maple cannot make this transformation in the "forward" direction, but can verify it "in reverse." Simply evaluate the derivative on the left to obtainexpand(convert(ode3,diff));which compares favorably with simplify(ode2);The equation ode3;can be solved by direct integration, that is, by antidifferentiation of both sides. The result isint_ode3 := map(Int, ode3, t );The left-hand side is trivial to evaluate, and Maple does a fine job with the right-hand side. After evaluating these indefinite integrals and adding the constant of integration, the result isq1 := subs( int_fact1, mu(t)*x(t) ) = int( rhs(ode3), t ) + C;The explicit general solution to the given first-order linear ODE is thereforeexpl_soln := op(solve( q1, {x(t)} ));That this solution satisfies the original differential equation is confirmed withodetest( expl_soln, lin_ode1 );To emphasize the structure of this solution, the homogeneous and particular solutions are respectivelysoln_h := x(t) = coeff( rhs(expl_soln), C );soln_p := x(t) = subs( C=0, rhs(expl_soln));as confirmed byodetest( soln_h, lhs(lin_ode1)=0 );odetest( soln_p, lin_ode1 );

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