75.18.10 problem 599

Internal problem ID [17098]
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. Initial value problem. Exercises page 140
Problem number : 599
Date solved : Tuesday, January 28, 2025 at 09:52:23 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

With initial conditions

\begin{align*} y \left (0\right )&=0\\ y^{\prime }\left (0\right )&=1 \end{align*}

✓ Solution by Maple

Time used: 0.017 (sec). Leaf size: 13

dsolve([diff(y(x),x$2)+y(x)=4*x*cos(x),y(0) = 0, D(y)(0) = 1],y(x), singsol=all)
 

\[ y = x \left (\cos \left (x \right )+x \sin \left (x \right )\right ) \]

✓ Solution by Mathematica

Time used: 0.077 (sec). Leaf size: 86

DSolve[{D[y[x],{x,2}]+y[x]==4*x*Cos[x],{y[0]==0,Derivative[1][y][0] ==1}},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \sin (x) \left (-\int _1^04 \cos ^2(K[2]) K[2]dK[2]\right )+\sin (x) \int _1^x4 \cos ^2(K[2]) K[2]dK[2]-\cos (x) \int _1^0-2 K[1] \sin (2 K[1])dK[1]+\cos (x) \int _1^x-2 K[1] \sin (2 K[1])dK[1]+\sin (x) \]