9.21 problem Problem 21

Internal problem ID [2285]

Book: Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section: Chapter 8, Linear differential equations of order n. Section 8.7, The Variation of Parameters Method. page 556
Problem number: Problem 21.
ODE order: 3.
ODE degree: 1.

CAS Maple gives this as type [[_3rd_order, _linear, _nonhomogeneous]]

Solve \begin {gather*} \boxed {y^{\prime \prime \prime }+3 y^{\prime \prime }+3 y^{\prime }+y-\frac {2 \,{\mathrm e}^{-x}}{x^{2}+1}=0} \end {gather*}

Solution by Maple

Time used: 0.015 (sec). Leaf size: 64

dsolve(diff(y(x),x$3)+3*diff(y(x),x$2)+3*diff(y(x),x)+y(x)=2*exp(-x)/(1+x^2),y(x), singsol=all)
 

\[ y \relax (x ) = \arctan \relax (x ) x^{2} {\mathrm e}^{-x}-\ln \left (x^{2}+1\right ) x \,{\mathrm e}^{-x}-{\mathrm e}^{-x} \arctan \relax (x )+x \,{\mathrm e}^{-x}+{\mathrm e}^{-x} c_{1}+c_{2} x \,{\mathrm e}^{-x}+c_{3} x^{2} {\mathrm e}^{-x} \]

Solution by Mathematica

Time used: 0.022 (sec). Leaf size: 40

DSolve[y'''[x]+3*y''[x]+3*y'[x]+y[x]==2*Exp[-x]/(1+x^2),y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to e^{-x} \left (\left (x^2-1\right ) \text {ArcTan}(x)+x \left (-\log \left (x^2+1\right )+c_3 x+c_2\right )+x+c_1\right ) \\ \end{align*}