12.9.8 problem 8

Internal problem ID [1764]
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
Date solved : Monday, January 27, 2025 at 05:34:34 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 x y^{\prime }+\left (4 x^{2}+2\right ) y&=8 \,{\mathrm e}^{-x \left (x +2\right )} \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&={\mathrm e}^{-x^{2}} \end{align*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 26

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)],singsol=all)
 

\[ y = \left (c_1 x +c_2 \right ) {\mathrm e}^{-x^{2}}+2 \,{\mathrm e}^{-x \left (x +2\right )} \]

✓ Solution by Mathematica

Time used: 0.051 (sec). Leaf size: 29

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

\[ y(x)\to e^{-x (x+2)} \left (2+e^{2 x} (c_2 x+c_1)\right ) \]