89.23.28 problem 32

Internal problem ID [24832]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 10. Nonhomogeneous Equations: Operational methods. Miscellaneous Exercises at page 162
Problem number : 32
Date solved : Thursday, October 02, 2025 at 10:48:25 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

With initial conditions

\begin{align*} y \left (0\right )&=0 \\ y^{\prime }\left (0\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.012 (sec). Leaf size: 15

ode:=diff(diff(y(x),x),x)-y(x) = sin(2*x); 
ic:=[y(0) = 0, D(y)(0) = 1]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = -\frac {\sin \left (2 x \right )}{5}+\frac {7 \sinh \left (x \right )}{5} \]

✓ Mathematica. Time used: 0.024 (sec). Leaf size: 28

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

\begin{align*} y(x)&\to \frac {1}{10} \left (-7 e^{-x}+7 e^x-4 \sin (x) \cos (x)\right ) \end{align*}

✓ Sympy. Time used: 0.063 (sec). Leaf size: 22

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-y(x) - sin(2*x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(x), x), x, 0): 1} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {7 e^{x}}{10} - \frac {\sin {\left (2 x \right )}}{5} - \frac {7 e^{- x}}{10} \]