75.17.24 problem 574

Internal problem ID [16994]
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 : 574
Date solved : Thursday, March 13, 2025 at 09:06:40 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 32

ode:=diff(diff(y(x),x),x)+y(x) = cos(2*x)^2+sin(1/2*x)^2; 
dsolve(ode,y(x), singsol=all);
 

\[ y = 1-\frac {\cos \left (4 x \right )}{30}+\frac {\left (-1+8 c_{1} \right ) \cos \left (x \right )}{8}+\frac {\left (-x +4 c_{2} \right ) \sin \left (x \right )}{4} \]

✓ Mathematica. Time used: 0.508 (sec). Leaf size: 74

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

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

✓ Sympy. Time used: 1.610 (sec). Leaf size: 46

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

\[ y{\left (x \right )} = C_{2} \cos {\left (x \right )} + \frac {\left (1 - \cos {\left (2 x \right )}\right )^{2}}{5} + \left (C_{1} - \frac {x}{4}\right ) \sin {\left (x \right )} + \frac {2 \cos {\left (2 x \right )}}{5} - \frac {2 \cos {\left (4 x \right )}}{15} + \frac {7}{10} \]