89.16.39 problem 39

Internal problem ID [24689]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 9. Nonhomogeneous Equations: Undetermined coefficients. Exercises at page 140
Problem number : 39
Date solved : Thursday, October 02, 2025 at 10:47:07 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

With initial conditions

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 10

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

\[ y = \sin \left (x \right ) \left (c_2 +x \right ) \]

✓ Mathematica. Time used: 0.041 (sec). Leaf size: 12

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

\begin{align*} y(x)&\to (x+c_2) \sin (x) \end{align*}

✓ Sympy. Time used: 0.051 (sec). Leaf size: 8

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

\[ y{\left (x \right )} = \left (C_{1} + x\right ) \sin {\left (x \right )} \]