64.16.10 problem 10

Internal problem ID [13542]
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 : 10
Date solved : Monday, March 31, 2025 at 08:00:51 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )-\frac {d}{d t}y \left (t \right )-2 x \left (t \right )+4 y \left (t \right )&=t\\ \frac {d}{d t}x \left (t \right )+\frac {d}{d t}y \left (t \right )-x \left (t \right )-y \left (t \right )&=1 \end{align*}

✓ Maple. Time used: 0.129 (sec). Leaf size: 39

ode:=[diff(x(t),t)-diff(y(t),t)-2*x(t)+4*y(t) = t, diff(x(t),t)+diff(y(t),t)-x(t)-y(t) = 1]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{t} c_2 +{\mathrm e}^{3 t} c_1 -\frac {t}{6}-\frac {13}{18} \\ y \left (t \right ) &= \frac {{\mathrm e}^{t} c_2}{3}-{\mathrm e}^{3 t} c_1 -\frac {5}{18}+\frac {t}{6} \\ \end{align*}

✓ Mathematica. Time used: 0.117 (sec). Leaf size: 227

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

\begin{align*} x(t)\to \frac {1}{4} e^t \left (\left (e^{2 t}+3\right ) \int _1^t\frac {1}{4} e^{-3 K[1]} \left (2 K[1]+3 e^{2 K[1]}-1\right )dK[1]-3 \left (e^{2 t}-1\right ) \int _1^t\frac {1}{4} e^{-3 K[2]} \left (-2 K[2]+e^{2 K[2]}+1\right )dK[2]+c_1 e^{2 t}-3 c_2 e^{2 t}+3 c_1+3 c_2\right ) \\ y(t)\to \frac {1}{4} e^t \left (-\left (e^{2 t}-1\right ) \int _1^t\frac {1}{4} e^{-3 K[1]} \left (2 K[1]+3 e^{2 K[1]}-1\right )dK[1]+\left (3 e^{2 t}+1\right ) \int _1^t\frac {1}{4} e^{-3 K[2]} \left (-2 K[2]+e^{2 K[2]}+1\right )dK[2]+c_1 \left (-e^{2 t}\right )+3 c_2 e^{2 t}+c_1+c_2\right ) \\ \end{align*}

✓ Sympy. Time used: 0.176 (sec). Leaf size: 42

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-t - 2*x(t) + 4*y(t) + Derivative(x(t), t) - Derivative(y(t), t),0),Eq(-x(t) - y(t) + Derivative(x(t), t) + Derivative(y(t), t) - 1,0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = 3 C_{1} e^{t} - C_{2} e^{3 t} - \frac {t}{6} - \frac {13}{18}, \ y{\left (t \right )} = C_{1} e^{t} + C_{2} e^{3 t} + \frac {t}{6} - \frac {5}{18}\right ] \]