75.25.1 problem 757

Internal problem ID [17224]
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 18.3. Finding periodic solutions of linear differential equations. Exercises page 187
Problem number : 757
Date solved : Tuesday, January 28, 2025 at 09:57:58 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 28

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

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

✓ Solution by Mathematica

Time used: 0.056 (sec). Leaf size: 55

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

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