75.20.33 problem 672

Internal problem ID [17167]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 15.5 Linear equations with variable coefficients. The Lagrange method. Exercises page 148
Problem number : 672
Date solved : Tuesday, January 28, 2025 at 09:55:51 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

With initial conditions

\begin{align*} y \left (-\infty \right )&=0\\ y^{\prime }\left (-1\right )&=-{\mathrm e}^{-1} \end{align*}

✓ Solution by Maple

Time used: 0.750 (sec). Leaf size: 10

dsolve([diff(y(x),x$2)+2/x*diff(y(x),x)-y(x)=4*exp(x),y(-infinity) = 0, D(y)(-1) = -1/exp(1)],y(x), singsol=all)
 

\[ y = \left (x -1\right ) {\mathrm e}^{x} \]

✓ Solution by Mathematica

Time used: 0.089 (sec). Leaf size: 12

DSolve[{D[y[x],{x,2}]+2/x*D[y[x],x]-y[x]==4*Exp[x],{y[-Infinity]==0,Derivative[1][y][-1]==-1/Exp[1]}},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to e^x (x-1) \]