85.83.9 problem 3

Internal problem ID [23006]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 10. Systems of differential equations and their applications. A Exercises at page 444
Problem number : 3
Date solved : Sunday, October 12, 2025 at 05:55:05 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d^{2}}{d t^{2}}x \left (t \right )+2 \frac {d}{d t}y \left (t \right )+8 x \left (t \right )&=32 t\\ \frac {d^{2}}{d t^{2}}y \left (t \right )+3 \frac {d}{d t}x \left (t \right )-2 y \left (t \right )&=60 \,{\mathrm e}^{-t} \end{align*}

With initial conditions

\begin{align*} x \left (0\right )&=6 \\ D\left (x \right )\left (0\right )&=0 \\ y \left (0\right )&=-24 \\ D\left (y \right )\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.280 (sec). Leaf size: 73

ode:=[diff(diff(x(t),t),t)+2*diff(y(t),t)+8*x(t) = 32*t, diff(diff(y(t),t),t)+3*diff(x(t),t)-2*y(t) = 60*exp(-t)]; 
ic:=[x(0) = 6, D(x)(0) = 0, y(0) = -24, D(y)(0) = 0]; 
dsolve([ode,op(ic)]);
 

\begin{align*} x \left (t \right ) &= 4 t -8 \,{\mathrm e}^{-t}-3 \sin \left (2 t \right )+12 \cos \left (2 t \right )+\frac {5 \,{\mathrm e}^{-2 t}}{2}-\frac {{\mathrm e}^{2 t}}{2} \\ y \left (t \right ) &= -36 \,{\mathrm e}^{-t}-3 \cos \left (2 t \right )-12 \sin \left (2 t \right )+\frac {15 \,{\mathrm e}^{-2 t}}{2}+\frac {3 \,{\mathrm e}^{2 t}}{2}+6 \\ \end{align*}

✓ Mathematica. Time used: 1.338 (sec). Leaf size: 78

ode={D[x[t],{t,2}]+2*D[y[t],t]+8*x[t]==32*t,D[y[t],{t,2}]+3*D[x[t],t]-2*y[t]==60*Exp[-t]}; 
ic={x[0]==0,Derivative[1][x][0] ==0,y[0]==-24,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to 4 t+4 e^{-2 t}-8 e^{-t}+e^{2 t}-3 \sin (2 t)+3 \cos (2 t)\\ y(t)&\to 12 e^{-2 t}-36 e^{-t}-3 e^{2 t}-3 \sin (2 t)-3 \cos (2 t)+6 \end{align*}

✓ Sympy. Time used: 0.676 (sec). Leaf size: 156

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-32*t + 8*x(t) + Derivative(x(t), (t, 2)) + 2*Derivative(y(t), t),0),Eq(-2*y(t) + 3*Derivative(x(t), t) + Derivative(y(t), (t, 2)) - 60*exp(-t),0)] 
ics = {x(0): 0, y(0): -24, Subs(Derivative(y(t), t), t, 0): 0, Subs(Derivative(x(t), t), t, 0): 0} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = 6 t \sin ^{2}{\left (2 t \right )} + 6 t \cos ^{2}{\left (2 t \right )} - 2 t + e^{2 t} - 3 \sin {\left (2 t \right )} + 3 \cos {\left (2 t \right )} - 3 e^{- t} \sin ^{2}{\left (2 t \right )} - 3 e^{- t} \cos ^{2}{\left (2 t \right )} - 5 e^{- t} + 4 e^{- 2 t}, \ y{\left (t \right )} = - 3 e^{2 t} + 3 \sin ^{2}{\left (2 t \right )} - 3 \sin {\left (2 t \right )} + 3 \cos ^{2}{\left (2 t \right )} - 3 \cos {\left (2 t \right )} + 3 - 6 e^{- t} \sin ^{2}{\left (2 t \right )} - 6 e^{- t} \cos ^{2}{\left (2 t \right )} - 30 e^{- t} + 12 e^{- 2 t}\right ] \]