75.17.12 problem 562

Internal problem ID [17061]
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 : 562
Date solved : Tuesday, January 28, 2025 at 09:48:08 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+2 y&=\left (5 x +4\right ) {\mathrm e}^{x}+{\mathrm e}^{-x} \end{align*}

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 29

dsolve(diff(y(x),x$2)+2*diff(y(x),x)+2*y(x)=(5*x+4)*exp(x)+exp(-x),y(x), singsol=all)
 

\[ y = \sin \left (x \right ) {\mathrm e}^{-x} c_{2} +{\mathrm e}^{-x} c_{1} \cos \left (x \right )+{\mathrm e}^{x} x +{\mathrm e}^{-x} \]

✓ Solution by Mathematica

Time used: 0.159 (sec). Leaf size: 81

DSolve[D[y[x],{x,2}]+2*D[y[x],x]+2*y[x]==(5*x+4)*Exp[x]+Exp[-x],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to e^{-x} \left (\cos (x) \int _1^x-\left (\left (e^{2 K[2]} (5 K[2]+4)+1\right ) \sin (K[2])\right )dK[2]+\sin (x) \int _1^x\cos (K[1]) \left (e^{2 K[1]} (5 K[1]+4)+1\right )dK[1]+c_2 \cos (x)+c_1 \sin (x)\right ) \]