69.25.1 problem 757

Internal problem ID [18511]
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 : Thursday, October 02, 2025 at 03:14:30 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

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

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

\[ y = \frac {\left (8 c_1 +1\right ) \cos \left (2 x \right )}{8}+\frac {1}{8}+\frac {\left (8 c_2 +x \right ) \sin \left (2 x \right )}{8} \]

✓ Mathematica. Time used: 0.033 (sec). Leaf size: 55

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

\begin{align*} 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) \end{align*}

✓ Sympy. Time used: 0.402 (sec). Leaf size: 22

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*y(x) - cos(x)**2 + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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