75.16.72 problem 545

Internal problem ID [17045]
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. Trial and error method. Exercises page 132
Problem number : 545
Date solved : Tuesday, January 28, 2025 at 09:47:30 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 30

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

\[ y = {\mathrm e}^{-x} \left (c_{2} +c_{1} x -\cos \left (x \right ) x^{2}+4 x \sin \left (x \right )+6 \cos \left (x \right )\right ) \]

✓ Solution by Mathematica

Time used: 0.031 (sec). Leaf size: 32

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

\[ y(x)\to e^{-x} \left (-\left (x^2-6\right ) \cos (x)+4 x \sin (x)+c_2 x+c_1\right ) \]