9.8 problem 8

Internal problem ID [1114]

Book: Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section: Chapter 5 linear second order equations. Section 5.6 Reduction or order. Page 253
Problem number: 8.
ODE order: 2.
ODE degree: 1.

CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]

Solve \begin {gather*} \boxed {y^{\prime \prime }+4 y^{\prime } x +\left (4 x^{2}+2\right ) y-8 \,{\mathrm e}^{-x \left (2+x \right )}=0} \end {gather*} Given that one solution of the ode is \begin {align*} y_1 &= {\mathrm e}^{-x^{2}} \end {align*}

Solution by Maple

Time used: 0.008 (sec). Leaf size: 31

dsolve([diff(y(x),x$2)+4*x*diff(y(x),x)+(4*x^2+2)*y(x)=8*exp(-x*(x+2)),exp(-x^2)],y(x), singsol=all)
 

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

Solution by Mathematica

Time used: 0.027 (sec). Leaf size: 29

DSolve[y''[x]+4*x*y'[x]+(4*x^2+2)*y[x]==8*Exp[-x*(x+2)],y[x],x,IncludeSingularSolutions -> True]
 

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