75.18.9 problem 598

Internal problem ID [17097]
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 : 598
Date solved : Tuesday, January 28, 2025 at 09:52:19 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

With initial conditions

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

✓ Solution by Maple

Time used: 0.086 (sec). Leaf size: 19

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

\[ y = \frac {\sin \left (2 x \right )}{3}+\cos \left (2 x \right )+\frac {\sin \left (x \right )}{3} \]

✓ Solution by Mathematica

Time used: 0.041 (sec). Leaf size: 76

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

\[ y(x)\to -\sin (2 x) \int _1^0\frac {1}{2} \cos (2 K[1]) \sin (K[1])dK[1]+\sin (2 x) \int _1^x\frac {1}{2} \cos (2 K[1]) \sin (K[1])dK[1]+\cos (2 x)-\frac {1}{3} \sin ^3(x) \cos (2 x)+\sin (x) \cos (x) \]