15.11.28 problem 28

Internal problem ID [3138]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 19, page 86
Problem number : 28
Date solved : Tuesday, March 04, 2025 at 04:02:55 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y&=12 \cos \left (x \right )^{2} \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.017 (sec). Leaf size: 25

ode:=diff(diff(y(x),x),x)+4*y(x) = 12*cos(x)^2; 
ic:=y(1/2*Pi) = 0, D(y)(1/2*Pi) = 1/2*Pi; 
dsolve([ode,ic],y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.032 (sec). Leaf size: 23

ode=D[y[x],{x,2}]+4*y[x]==12*Cos[x]^2; 
ic={y[Pi/2]==0,Derivative[1][y][Pi/2]==Pi/2}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \cos (x) ((3 x-2 \pi ) \sin (x)+3 \cos (x)) \]

✓ Sympy. Time used: 0.688 (sec). Leaf size: 26

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*y(x) - 12*cos(x)**2 + Derivative(y(x), (x, 2)),0) 
ics = {y(pi/2): 0, Subs(Derivative(y(x), x), x, pi/2): pi/2} 
dsolve(ode,func=y(x),ics=ics)
 

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