75.17.23 problem 573

Internal problem ID [17072]
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 15.3 Nonhomogeneous linear equations with constant coefficients. Superposition principle. Exercises page 137
Problem number : 573
Date solved : Tuesday, January 28, 2025 at 09:49:22 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+3 y^{\prime }+2 y&=6 x \,{\mathrm e}^{-x} \left (1-{\mathrm e}^{-x}\right ) \end{align*}

Solution by Maple

Time used: 0.003 (sec). Leaf size: 36

dsolve(diff(y(x),x$2)+3*diff(y(x),x)+2*y(x)=6*x*exp(-x)*(1-exp(-x)),y(x), singsol=all)
 
\[ y = 3 \,{\mathrm e}^{-x} \left (\left (x^{2}+2 x -\frac {1}{3} c_{1} +2\right ) {\mathrm e}^{-x}+x^{2}-2 x +\frac {c_{2}}{3}\right ) \]

Solution by Mathematica

Time used: 0.323 (sec). Leaf size: 62

DSolve[D[y[x],{x,2}]+3*D[y[x],x]+2*y[x]==6*x*Exp[-x]*(1-Exp[-x]),y[x],x,IncludeSingularSolutions -> True]
 
\[ y(x)\to e^{-2 x} \left (\int _1^x-6 \left (-1+e^{K[1]}\right ) K[1]dK[1]+e^x \int _1^x6 \left (1-e^{-K[2]}\right ) K[2]dK[2]+c_2 e^x+c_1\right ) \]