87.17.49 problem 58

Internal problem ID [23641]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 2. Linear differential equations. Exercise at page 127
Problem number : 58
Date solved : Thursday, October 02, 2025 at 09:43:43 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

With initial conditions

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

✓ Maple. Time used: 0.074 (sec). Leaf size: 41

ode:=diff(diff(y(x),x),x)+y(x) = sin(a*x); 
ic:=[y(0) = y__0, D(y)(0) = y__1]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {\sin \left (x \right ) \left (a^{2} y_{1} +a -y_{1} \right )+\cos \left (x \right ) y_{0} \left (a^{2}-1\right )-\sin \left (a x \right )}{a^{2}-1} \]

✓ Mathematica. Time used: 0.093 (sec). Leaf size: 42

ode=D[y[x],{x,2}]+y[x]==Sin[a*x]; 
ic={y[0]==y0,Derivative[1][y][0] ==y1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {\left (a^2-1\right ) \text {y0} \cos (x)+\left (a^2 \text {y1}+a-\text {y1}\right ) \sin (x)-\sin (a x)}{a^2-1} \end{align*}

✓ Sympy. Time used: 0.067 (sec). Leaf size: 34

from sympy import * 
x = symbols("x") 
y0 = symbols("y0") 
y1 = symbols("y1") 
y = Function("y") 
ode = Eq(y(x) - sin(a*x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): y0, Subs(Derivative(y(x), x), x, 0): y1} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = y_{0} \cos {\left (x \right )} + \frac {\left (a^{2} y_{1} + a - y_{1}\right ) \sin {\left (x \right )}}{a^{2} - 1} - \frac {\sin {\left (a x \right )}}{a^{2} - 1} \]