85.87.3 problem 5

Internal problem ID [23027]
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. B Exercises at page 491
Problem number : 5
Date solved : Sunday, October 12, 2025 at 05:55:07 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d^{2}}{d t^{2}}x \left (t \right )&=y \left (t \right )+4 \,{\mathrm e}^{-2 t}\\ \frac {d^{2}}{d t^{2}}y \left (t \right )&=x \left (t \right )-{\mathrm e}^{-2 t} \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.101 (sec). Leaf size: 47

ode:=[diff(diff(x(t),t),t) = y(t)+4*exp(-2*t), diff(diff(y(t),t),t) = x(t)-exp(-2*t)]; 
ic:=[x(0) = 0, D(x)(0) = 0, y(0) = 0, D(y)(0) = 0]; 
dsolve([ode,op(ic)]);
 

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

✓ Mathematica. Time used: 0.135 (sec). Leaf size: 61

ode={D[x[t],{t,2}]==y[t]+4*Exp[-2*t],D[y[t],{t,2}]==x[t]-Exp[-2*t]}; 
ic={x[0]==0,Derivative[1][x][0] ==0,y[0]==0,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.398 (sec). Leaf size: 109

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-y(t) + Derivative(x(t), (t, 2)) - 4*exp(-2*t),0),Eq(-x(t) + Derivative(y(t), (t, 2)) + exp(-2*t),0)] 
ics = {x(0): 0, Subs(Derivative(x(t), t), t, 0): 0, y(0): 0, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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