75.17.21 problem 571

Internal problem ID [17070]
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 : 571
Date solved : Tuesday, January 28, 2025 at 09:49:15 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-2 y^{\prime }-3 y&=2 x +{\mathrm e}^{-x}-2 \,{\mathrm e}^{3 x} \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 37

dsolve(diff(y(x),x$2)-2*diff(y(x),x)-3*y(x)=2*x+exp(-x)-2*exp(3*x),y(x), singsol=all)
 

\[ y = \frac {4}{9}+\frac {\left (-1-4 x +16 c_{1} \right ) {\mathrm e}^{-x}}{16}+\frac {\left (1-4 x +8 c_{2} \right ) {\mathrm e}^{3 x}}{8}-\frac {2 x}{3} \]

✓ Solution by Mathematica

Time used: 0.591 (sec). Leaf size: 86

DSolve[D[y[x],{x,2}]-2*D[y[x],x]-3*y[x]==2*x+Exp[-x]-2*Exp[3*x],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to e^{-x} \left (\int _1^x\frac {1}{4} \left (-2 e^{K[1]} K[1]+2 e^{4 K[1]}-1\right )dK[1]+e^{4 x} \int _1^x\frac {1}{4} \left (2 e^{-3 K[2]} K[2]+e^{-4 K[2]}-2\right )dK[2]+c_2 e^{4 x}+c_1\right ) \]