56.29.8 problem Ex 9

Internal problem ID [14240]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter VII, Linear differential equations with constant coefficients. Article 52. Summary. Page 117
Problem number : Ex 9
Date solved : Thursday, October 02, 2025 at 09:27:04 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 30

ode:=diff(diff(y(x),x),x)+4*y(x) = sin(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: 0.059 (sec). Leaf size: 34

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

\begin{align*} y(x)&\to \frac {1}{8} ((-1+8 c_1) \cos (2 x)-(x-8 c_2) \sin (2 x)+1) \end{align*}

✓ Sympy. Time used: 0.411 (sec). Leaf size: 22

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

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