85.50.1 problem 1

Internal problem ID [22810]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 4. Linear differential equations. B Exercises at page 195
Problem number : 1
Date solved : Thursday, October 02, 2025 at 09:14:49 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y&=6 \cos \left (x \right )^{2} \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=0 \\ y \left (\frac {\pi }{2}\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.012 (sec). Leaf size: 20

ode:=diff(diff(y(x),x),x)+y(x) = 6*cos(x)^2; 
ic:=[y(0) = 0, y(1/2*Pi) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = -4 \sin \left (x \right )-2 \cos \left (x \right )+3-\cos \left (2 x \right ) \]

✓ Mathematica. Time used: 0.02 (sec). Leaf size: 23

ode=D[y[x],{x,2}]+y[x]==8*Cos[x]^2; 
ic={y[0]==0,y[Pi/2]==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {4}{3} (4 \sin (x)+2 \cos (x)+\cos (2 x)-3) \end{align*}

✓ Sympy. Time used: 0.205 (sec). Leaf size: 26

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(y(x) - 8*cos(x)**2 + Derivative(y(x), (x, 2)),0) 
ics = {y(0): 0, y(pi/2): 0} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = - \frac {16 \sin {\left (x \right )}}{3} - \frac {8 \cos {\left (x \right )}}{3} - \frac {4 \cos {\left (2 x \right )}}{3} + 4 \]