64.16.16 problem 16

Internal problem ID [13548]
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 : 16
Date solved : Monday, March 31, 2025 at 08:01:02 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.182 (sec). Leaf size: 44

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

\begin{align*} x \left (t \right ) &= \frac {t^{2}}{2}+\frac {3 t}{2}+\frac {3}{4}+{\mathrm e}^{2 t} c_2 \\ y \left (t \right ) &= \frac {15}{4}+\frac {3 t}{2}-\frac {t^{2}}{2}-3 \,{\mathrm e}^{2 t} c_2 +c_1 \,{\mathrm e}^{t} \\ \end{align*}

✓ Mathematica. Time used: 0.282 (sec). Leaf size: 115

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

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

✓ Sympy. Time used: 0.173 (sec). Leaf size: 51

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

\[ \left [ x{\left (t \right )} = - \frac {C_{1} e^{2 t}}{3} + \frac {t^{2}}{2} + \frac {3 t}{2} + \frac {3}{4}, \ y{\left (t \right )} = C_{1} e^{2 t} + C_{2} e^{t} - \frac {t^{2}}{2} + \frac {3 t}{2} + \frac {15}{4}\right ] \]