85.57.2 problem 2

Internal problem ID [22837]
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 200
Problem number : 2
Date solved : Thursday, October 02, 2025 at 09:15:34 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

With initial conditions

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

✓ Maple. Time used: 0.020 (sec). Leaf size: 40

ode:=diff(diff(y(t),t),t)+omega^2*y(t) = t*(sin(omega*t)+cos(omega*t)); 
ic:=[y(0) = 0, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = \frac {\left (\omega ^{2} t^{2}+\omega t -1\right ) \sin \left (\omega t \right )-\cos \left (\omega t \right ) \omega t \left (\omega t -1\right )}{4 \omega ^{3}} \]

✓ Mathematica. Time used: 0.163 (sec). Leaf size: 43

ode=D[y[t],{t,2}]+\[Omega]^2*y[t]==t*(Sin[\[Omega]*t]+Cos[\[Omega]*t]); 
ic={y[0]==0,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {\left (t^2 \omega ^2+t \omega -1\right ) \sin (t \omega )-t \omega (t \omega -1) \cos (t \omega )}{4 \omega ^3} \end{align*}

✓ Sympy. Time used: 0.190 (sec). Leaf size: 76

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

\[ y{\left (t \right )} = \frac {t^{2} \sin {\left (t w \right )}}{4 w} - \frac {t^{2} \cos {\left (t w \right )}}{4 w} + \frac {t \sin {\left (t w \right )}}{4 w^{2}} + \frac {t \cos {\left (t w \right )}}{4 w^{2}} + \frac {i e^{i t w}}{8 w^{3}} - \frac {i e^{- i t w}}{8 w^{3}} \]