85.86.4 problem 1 (d)

Internal problem ID [23021]
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 491
Problem number : 1 (d)
Date solved : Thursday, October 02, 2025 at 09:17:34 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )-3 x \left (t \right )-6 y \left (t \right )&=27 t^{2}\\ \frac {d}{d t}x \left (t \right )+\frac {d}{d t}y \left (t \right )-3 y \left (t \right )&=5 \,{\mathrm e}^{t} \end{align*}

With initial conditions

\begin{align*} x \left (0\right )&=5 \\ y \left (0\right )&=-1 \\ \end{align*}

✓ Maple. Time used: 0.258 (sec). Leaf size: 29

ode:=[diff(x(t),t)-3*x(t)-6*y(t) = 27*t^2, diff(x(t),t)+diff(y(t),t)-3*y(t) = 5*exp(t)]; 
ic:=[x(0) = 5, y(0) = -1]; 
dsolve([ode,op(ic)]);
 

\begin{align*} x \left (t \right ) &= -9 t^{2}+3 \,{\mathrm e}^{t}+2+6 t \\ y \left (t \right ) &= -{\mathrm e}^{t}-6 t \\ \end{align*}

✓ Mathematica. Time used: 0.124 (sec). Leaf size: 32

ode={D[x[t],{t,1}]-3*x[t]-6*y[t]==27*t^2,D[x[t],t]+ D[y[t],t]-3*y[t]==5*Exp[t]}; 
ic={x[0]==5,y[0]==-1}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to -9 t^2+6 t+3 e^t+2\\ y(t)&\to -6 t-e^t \end{align*}

✓ Sympy. Time used: 0.319 (sec). Leaf size: 131

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-27*t**2 - 3*x(t) - 6*y(t) + Derivative(x(t), t),0),Eq(-3*y(t) - 5*exp(t) + Derivative(x(t), t) + Derivative(y(t), t),0)] 
ics = {x(0): 5, y(0): -1} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - 9 t^{2} \sin ^{2}{\left (3 t \right )} - 9 t^{2} \cos ^{2}{\left (3 t \right )} + 6 t \sin ^{2}{\left (3 t \right )} + 6 t \cos ^{2}{\left (3 t \right )} + 3 e^{t} \sin ^{2}{\left (3 t \right )} + 3 e^{t} \cos ^{2}{\left (3 t \right )} + 2 \sin ^{2}{\left (3 t \right )} + 2 \cos ^{2}{\left (3 t \right )}, \ y{\left (t \right )} = - 6 t \sin ^{2}{\left (3 t \right )} - 6 t \cos ^{2}{\left (3 t \right )} - e^{t} \sin ^{2}{\left (3 t \right )} - e^{t} \cos ^{2}{\left (3 t \right )}\right ] \]