12.10.10 problem 10

Internal problem ID [1814]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 5 linear second order equations. Section 5.7 Variation of Parameters. Page 262
Problem number : 10
Date solved : Monday, January 27, 2025 at 05:35:50 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 24

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

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

✓ Solution by Mathematica

Time used: 0.035 (sec). Leaf size: 27

DSolve[4*x^2*D[y[x],{x,2}]+(4*x-8*x^2)*D[y[x],x]+(4*x^2-4*x-1)*y[x]==4*x^(1/2)*Exp[x],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {e^x (x \log (x)+(-1+c_2) x+c_1)}{\sqrt {x}} \]