64.16.7 problem 7

Internal problem ID [13539]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 7, Systems of linear differential equations. Section 7.1. Exercises page 277
Problem number : 7
Date solved : Monday, March 31, 2025 at 08:00:46 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.167 (sec). Leaf size: 65

ode:=[diff(x(t),t)+diff(y(t),t)-x(t)-6*y(t) = exp(3*t), diff(x(t),t)+2*diff(y(t),t)-2*x(t)-6*y(t) = t]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{\sqrt {6}\, t} c_2 +{\mathrm e}^{-\sqrt {6}\, t} c_1 -t +\frac {1}{6} \\ y \left (t \right ) &= \frac {\sqrt {6}\, {\mathrm e}^{\sqrt {6}\, t} c_2}{6}-\frac {\sqrt {6}\, {\mathrm e}^{-\sqrt {6}\, t} c_1}{6}-\frac {1}{6}+\frac {t}{6}-\frac {{\mathrm e}^{3 t}}{3} \\ \end{align*}

✓ Mathematica. Time used: 7.401 (sec). Leaf size: 142

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

\begin{align*} x(t)\to \frac {1}{6} \left (-6 t+3 \left (c_1-\sqrt {6} c_2\right ) e^{-\sqrt {6} t}+3 \left (c_1+\sqrt {6} c_2\right ) e^{\sqrt {6} t}+1\right ) \\ y(t)\to \frac {1}{12} e^{-\sqrt {6} t} \left (2 e^{\sqrt {6} t} (t-1)-4 e^{\left (3+\sqrt {6}\right ) t}+\left (\sqrt {6} c_1+6 c_2\right ) e^{2 \sqrt {6} t}-\sqrt {6} c_1+6 c_2\right ) \\ \end{align*}

✓ Sympy. Time used: 0.433 (sec). Leaf size: 73

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

\[ \left [ x{\left (t \right )} = \sqrt {6} C_{1} e^{\sqrt {6} t} - \sqrt {6} C_{2} e^{- \sqrt {6} t} - t + \frac {1}{6}, \ y{\left (t \right )} = C_{1} e^{\sqrt {6} t} + C_{2} e^{- \sqrt {6} t} + \frac {t}{6} - \frac {e^{3 t}}{3} - \frac {1}{6}\right ] \]