dsolve - Maple Help

All Products Maple MapleSim

Home : Support : Online Help : Mathematics : Differential Equations : Example Worksheets : dsolve

Closed-form Solutions of Linear Differential Equations

The Maple dsolve command allows determination of closed-form solutions for linear differential equations.

Second-order Equations

The dsolve command converts a homogeneous second-order linear differential equation of the form

$ode := a (x) (\frac{{&DifferentialD;}^{2}}{&DifferentialD; x^{2}} y (x)) + b (x) (\frac{&DifferentialD;}{&DifferentialD; x} y (x)) + c (x) y (x) = 0$ into one of the form

$new_ode := \frac{{&DifferentialD;}^{2}}{&DifferentialD; x^{2}} y (x) + I (x) y (x) = 0$ , and then uses solutions of the new equation to build solutions to the first. The method computes a solution to the second by looking at the partial fraction expansion of I(x).

For example, we have (from the classical text by Kamke):

$restart$

>	$ode1 := \frac{{&DifferentialD;}^{2}}{&DifferentialD; x^{2}} y (x) + 2 x^{2} (\frac{&DifferentialD;}{&DifferentialD; x} y (x)) + 2 x y (x)$

$ode1 := \frac{{&DifferentialD;}^{2}}{&DifferentialD; x^{2}} y (x) + 2 x^{2} (\frac{&DifferentialD;}{&DifferentialD; x} y (x)) + 2 x y (x)$

(1.1)

>	$ans1 := dsolve (ode1, y (x))$

$ans1 := y (x) =_C1 {&ExponentialE;}^{- \frac{1}{3} x^{3}} \sqrt{x} BesselI (\frac{1}{6}, \frac{1}{3} x^{3}) +_C2 {&ExponentialE;}^{- \frac{1}{3} x^{3}} \sqrt{x} BesselK (\frac{1}{6}, \frac{1}{3} x^{3})$

(1.2)

>	$ode2 := (4 x^{4} - 1) (\frac{&DifferentialD;}{&DifferentialD; x} (\frac{&DifferentialD;}{&DifferentialD; x} y (x))) + \frac{(12 x^{4} + 1) (\frac{&DifferentialD;}{&DifferentialD; x} y (x))}{x} + 16 x^{2} (- \frac{u^{2}}{4 x^{4} - 1} - v (v + 1)) y (x)$

$ode2 := (4 x^{4} - 1) (\frac{{&DifferentialD;}^{2}}{&DifferentialD; x^{2}} y (x)) + \frac{(12 x^{4} + 1) (\frac{&DifferentialD;}{&DifferentialD; x} y (x))}{x} + 16 x^{2} (- \frac{u^{2}}{4 x^{4} - 1} - v (v + 1)) y (x)$

(1.3)

>	$ans2 := dsolve (ode2, y (x))$

$ans2 := y (x) =_C1 LegendreP (v, u, 2 x^{2}) +_C2 LegendreQ (v, u, 2 x^{2})$

(1.4)

>	$ode3 := \frac{{&DifferentialD;}^{2}}{&DifferentialD; x^{2}} y (x) - x (\frac{&DifferentialD;}{&DifferentialD; x} y (x)) + n y (x)$

$ode3 := \frac{{&DifferentialD;}^{2}}{&DifferentialD; x^{2}} y (x) - x (\frac{&DifferentialD;}{&DifferentialD; x} y (x)) + n y (x)$

(1.5)

>	$ans3 := dsolve (ode3, y (x))$

$ans3 := y (x) =_C1 KummerM (\frac{1}{2} - \frac{1}{2} n, \frac{3}{2}, \frac{1}{2} x^{2}) x +_C2 KummerU (\frac{1}{2} - \frac{1}{2} n, \frac{3}{2}, \frac{1}{2} x^{2}) x$

(1.6)

>	$ode4 := (x - 9 x^{5}) (\frac{{&DifferentialD;}^{2}}{&DifferentialD; x^{2}} y (x)) + (1 - 9 x^{4}) (\frac{&DifferentialD;}{&DifferentialD; x} y (x)) + 36 x^{3} y (x)$

$ode4 := (x - 9 x^{5}) (\frac{{&DifferentialD;}^{2}}{&DifferentialD; x^{2}} y (x)) + (1 - 9 x^{4}) (\frac{&DifferentialD;}{&DifferentialD; x} y (x)) + 36 x^{3} y (x)$

(1.7)

>	$ans4 := dsolve (ode4, y (x))$

$ans4 := y (x) =_C1 EllipticE (3 x^{2}) +_C2 (EllipticCE (3 x^{2}) - EllipticCK (3 x^{2}))$

(1.8)

In all of these examples, one can verify the solutions by using the odetest command:

>	$odetest (ans1, ode1)$

$0$

(1.9)

>	$odetest (ans2, ode2)$

$0$

(1.10)

>	$odetest (ans3, ode3)$

$0$

(1.11)

>	$odetest (ans4, ode4)$

$0$

(1.12)

Higher-order Equations

One can also solve higher-order equations. For example, in the equation

$ode := - 300 y (x) + 192 x (\frac{&DifferentialD;}{&DifferentialD; x} y (x)) + (- 47 x^{2} + 4 x^{3} μ - x^{4} - 4 x^{2} ν^{2}) (\frac{{&DifferentialD;}^{2}}{&DifferentialD; x^{2}} y (x)) + 12 y (x) x^{2} - 48 y (x) μ x + 48 y (x) ν^{2} + 4 x^{4} (\frac{{&DifferentialD;}^{4}}{&DifferentialD; x^{4}} y (x))$

(2.1)

two solutions are determined by the rational function solver DEtools[ratsols]. Reduction of order then produces a final answer in terms of integrals:

>	$dsolve (ode, y (x))$

$y (x) = \frac{_C1}{x^{3}} +_C2 x^{4} + \frac{_C3 (- (\int x^{4} WhittakerM (μ, ν, x) &DifferentialD; x) + (\int \frac{WhittakerM (μ, ν, x)}{x^{3}} &DifferentialD; x) x^{7})}{x^{3}} + \frac{_C4 (\int x^{4} WhittakerW (μ, ν, x) &DifferentialD; x - (\int \frac{WhittakerW (μ, ν, x)}{x^{3}} &DifferentialD; x) x^{7})}{x^{3}}$

(2.2)

>	$odetest (, ode)$

$0$

(2.3)

Maple does a quick test to determine if a particular ODE is the symmetric product of a second-order equation. If so, then a solution can be determined from the solutions of the second-order equation. For example, the equation

>	$ode := \frac{{&DifferentialD;}^{3}}{&DifferentialD; x^{3}} y (x) - 4 x (\frac{&DifferentialD;}{&DifferentialD; x} y (x)) - 2 y (x)$

$ode := \frac{{&DifferentialD;}^{3}}{&DifferentialD; x^{3}} y (x) - 4 x (\frac{&DifferentialD;}{&DifferentialD; x} y (x)) - 2 y (x)$

(2.4)

found in Kamke or Abramowitz and Stegun is the symmetric power of Airy's equation. As such, dsolve produces:

>	$dsolve (ode, y (x))$

$y (x) =_C1 {AiryBi (x)}^{2} +_C2 {AiryAi (x)}^{2} +_C3 AiryAi (x) AiryBi (x)$

(2.5)

Similarly,

>	$ode2 := \frac{{&DifferentialD;}^{3}}{&DifferentialD; x^{3}} y (x) - 6 x (\frac{{&DifferentialD;}^{2}}{&DifferentialD; x^{2}} y (x)) + 2 (4 x^{2} + 2 a - 1) (\frac{&DifferentialD;}{&DifferentialD; x} y (x)) - 8 a x y (x)$

$ode2 := \frac{{&DifferentialD;}^{3}}{&DifferentialD; x^{3}} y (x) - 6 x (\frac{{&DifferentialD;}^{2}}{&DifferentialD; x^{2}} y (x)) + 2 (4 x^{2} + 2 a - 1) (\frac{&DifferentialD;}{&DifferentialD; x} y (x)) - 8 a x y (x)$

(2.6)

>	$ans2 := dsolve (ode2, y (x))$

$ans2 := y (x) =_C1 {KummerU (\frac{1}{2} - \frac{1}{4} a, \frac{3}{2}, x^{2})}^{2} x^{2} +_C2 {KummerM (\frac{1}{2} - \frac{1}{4} a, \frac{3}{2}, x^{2})}^{2} x^{2} +_C3 KummerM (\frac{1}{2} - \frac{1}{4} a, \frac{3}{2}, x^{2}) x^{2} KummerU (\frac{1}{2} - \frac{1}{4} a, \frac{3}{2}, x^{2})$

(2.7)

>	$odetest (ans2, ode2)$

$0$

(2.8)

Combined with previous methods, we find that the equation

>	$ode := 2 y (x) + (- 2 + 4 x) (\frac{&DifferentialD;}{&DifferentialD; x} y (x)) - 4 x (\frac{{&DifferentialD;}^{2}}{&DifferentialD; x^{2}} y (x)) - (\frac{{&DifferentialD;}^{3}}{&DifferentialD; x^{3}} y (x)) + \frac{{&DifferentialD;}^{4}}{&DifferentialD; x^{4}} y (x)$

$ode := 2 y (x) + (- 2 + 4 x) (\frac{&DifferentialD;}{&DifferentialD; x} y (x)) - 4 x (\frac{{&DifferentialD;}^{2}}{&DifferentialD; x^{2}} y (x)) - (\frac{{&DifferentialD;}^{3}}{&DifferentialD; x^{3}} y (x)) + \frac{{&DifferentialD;}^{4}}{&DifferentialD; x^{4}} y (x)$

(2.9)

has a single answer determined from the Maple exponential solver DEtools[expsols], and then reduction of order reduces to a symmetric equation. This gives

>	$dsolve (ode, y (x))$

$y (x) =_C1 {&ExponentialE;}^{x} +_C2 (\int {AiryAi (x)}^{2} {&ExponentialE;}^{- x} &DifferentialD; x) {&ExponentialE;}^{x} +_C3 (\int {AiryBi (x)}^{2} {&ExponentialE;}^{- x} &DifferentialD; x) {&ExponentialE;}^{x} +_C4 (\int AiryAi (x) AiryBi (x) {&ExponentialE;}^{- x} &DifferentialD; x) {&ExponentialE;}^{x}$

(2.10)

>	$odetest (, ode)$

$0$

(2.11)

Additional Improvements

Improvements to solving linear ODEs also naturally lead to improvements in solving other differential equations. For example, first-order Riccati equations are typically converted to second-order linear equations to build their solutions. Similarly, linear systems of first-order equations are solved by building one or more higher-order scalar equations and by constructing the matrix solutions. A simple example is found by

>	$ode := \frac{&DifferentialD;}{&DifferentialD; x} y (x) = x + y (x) + a {y (x)}^{2}$

$ode := \frac{&DifferentialD;}{&DifferentialD; x} y (x) = x + y (x) + a {y (x)}^{2}$

(3.1)

>	$dsolve (ode, y (x))$

$y (x) = \frac{1}{2} \frac{-_C1 AiryAi (- \frac{1}{4} \frac{- 1 + 4 a x}{a^{2 / 3}}) + 2_C1 AiryAi (1, - \frac{1}{4} \frac{- 1 + 4 a x}{a^{2 / 3}}) a^{1 / 3} - AiryBi (- \frac{1}{4} \frac{- 1 + 4 a x}{a^{2 / 3}}) + 2 AiryBi (1, - \frac{1}{4} \frac{- 1 + 4 a x}{a^{2 / 3}}) a^{1 / 3}}{a (_C1 AiryAi (- \frac{1}{4} \frac{- 1 + 4 a x}{a^{2 / 3}}) + AiryBi (- \frac{1}{4} \frac{- 1 + 4 a x}{a^{2 / 3}}))}$

(3.2)

In the above answer, the functions AiryAi(1, .. ) and AiryBi(1,...) represent the derivatives of the AiryAi and AiryBi functions, respectively.

Return to Index for Example Worksheets

Download Help Document

What kind of issue would you like to report? (Optional)
Please add your Comment (Optional)
E-mail Address (Optional)
What is ?	This question helps us to combat spam

Maple

Maple Add-Ons

MapleSim

MapleSim Add-Ons

Systems Engineering

Consulting Services

Maple T.A. and Möbius

Education

Industries

Automotive and Aerospace

Robotics

Machine Design & Industrial Automation

Other

Application Areas

Product Pricing

Purchasing

Institutional Student Licensing

Maplesoft Elite Maintenance (EMP)

Support

Product Training

Online Product Help

Webinars & Events

Publications

Content Hubs

Examples & Applications

Community

About Maplesoft

Media Center

User Community

Contact

Online Help

All Products Maple MapleSim

Was this information helpful?

Tell us what we can do better:

Thanks for your Comment

Maple

Powerful math software that is easy to use

Maple Add-Ons

MapleSim

Advanced System Level Modeling

MapleSim Add-Ons

Systems Engineering

Consulting Services

Maple T.A. and Möbius

Education

Industries

Automotive and Aerospace

Robotics

Machine Design & Industrial Automation

Other

Application Areas

Product Pricing

Purchasing

Institutional Student Licensing

Maplesoft Elite Maintenance (EMP)

Support

Product Training

Online Product Help

Webinars & Events

Publications

Content Hubs

Examples & Applications

Community

About Maplesoft

Media Center

User Community

Contact

Online Help

All Products Maple MapleSim

Was this information helpful?

Tell us what we can do better:

Thanks for your Comment