84.28.6 problem 16.10

Internal problem ID [22283]
Book : Schaums outline series. Differential Equations By Richard Bronson. 1973. McGraw-Hill Inc. ISBN 0-07-008009-7
Section : Chapter 16. Initial-value problems. Supplementary problems
Problem number : 16.10
Date solved : Thursday, October 02, 2025 at 08:37:04 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

With initial conditions

\begin{align*} y \left (\pi \right )&=0 \\ y^{\prime }\left (\pi \right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.009 (sec). Leaf size: 14

ode:=diff(diff(y(x),x),x)+4*y(x) = sin(2*x)^2; 
ic:=[y(Pi) = 0, D(y)(Pi) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {\left (-1+\cos \left (2 x \right )\right )^{2}}{12} \]

✓ Mathematica. Time used: 0.016 (sec). Leaf size: 13

ode=D[y[x],{x,2}]+4*y[x]==Sin[2*x]^2; 
ic={y[Pi]==0,Derivative[1][y][Pi] ==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {\sin ^4(x)}{3} \end{align*}

✓ Sympy. Time used: 0.265 (sec). Leaf size: 26

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

\[ y{\left (x \right )} = \frac {\sin ^{4}{\left (x \right )}}{3} - \frac {\sin ^{2}{\left (x \right )}}{3} - \frac {\cos {\left (2 x \right )}}{6} + \frac {1}{6} \]