75.17.15 problem 565

Internal problem ID [17064]
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 : 565
Date solved : Tuesday, January 28, 2025 at 09:48:44 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 37

dsolve(diff(y(x),x$2)+4*y(x)=x*sin(x)^2,y(x), singsol=all)
 

\[ y = \frac {\left (-8 x^{2}+128 c_{2} +1\right ) \sin \left (2 x \right )}{128}+\frac {\left (-x +32 c_{1} \right ) \cos \left (2 x \right )}{32}+\frac {x}{8} \]

✓ Solution by Mathematica

Time used: 0.149 (sec). Leaf size: 72

DSolve[D[y[x],{x,2}]+4*y[x]==x*Sin[x]^2,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \cos (2 x) \int _1^x-\cos (K[1]) K[1] \sin ^3(K[1])dK[1]+\sin (2 x) \int _1^x\frac {1}{2} \cos (2 K[2]) K[2] \sin ^2(K[2])dK[2]+c_1 \cos (2 x)+c_2 \sin (2 x) \]