8.11.32 problem 55

Internal problem ID [900]
Book : Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section : Section 5.5, Nonhomogeneous equations and undetermined coefficients Page 351
Problem number : 55
Date solved : Tuesday, March 04, 2025 at 12:00:10 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 30

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

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

✓ Mathematica. Time used: 0.058 (sec). Leaf size: 71

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

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

✓ Sympy. Time used: 0.642 (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)
 

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