81.14.23 problem 18-36

Internal problem ID [21708]
Book : The Differential Equations Problem Solver. VOL. I. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 18. Algebra of differential operators. Page 435
Problem number : 18-36
Date solved : Thursday, October 02, 2025 at 08:00:11 PM
CAS classification : system_of_ODEs

\begin{align*} 2 \frac {d}{d t}x \left (t \right )-3 x \left (t \right )-2 \frac {d}{d t}y \left (t \right )&=t\\ 2 \frac {d}{d t}x \left (t \right )+3 x \left (t \right )+2 \frac {d}{d t}y \left (t \right )+8 y \left (t \right )&=2 \end{align*}

✓ Maple. Time used: 0.045 (sec). Leaf size: 39

ode:=[2*diff(x(t),t)-3*x(t)-2*diff(y(t),t) = t, 2*diff(x(t),t)+3*x(t)+2*diff(y(t),t)+8*y(t) = 2]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-3 t} c_2 +{\mathrm e}^{t} c_1 -\frac {t}{3}-\frac {11}{36} \\ y \left (t \right ) &= \frac {3 \,{\mathrm e}^{-3 t} c_2}{2}-\frac {{\mathrm e}^{t} c_1}{2}+\frac {5}{12}+\frac {t}{8} \\ \end{align*}

✓ Mathematica. Time used: 0.147 (sec). Leaf size: 81

ode={2*D[x[t],t]-3*x[t]-2*D[y[t],t]==t, 2*D[x[t],t]+3*x[t]+2*D[y[t],t]+8*y[t]==2 }; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{36} \left (-12 t+9 (3 c_1-2 c_2) e^t+9 (c_1+2 c_2) e^{-3 t}-11\right )\\ y(t)&\to \frac {1}{24} \left (3 t+9 (c_1+2 c_2) e^{-3 t}+(6 c_2-9 c_1) e^t+10\right ) \end{align*}

✓ Sympy. Time used: 0.425 (sec). Leaf size: 100

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-t - 3*x(t) + Derivative(x(t), t) - 2*Derivative(y(t), t),0),Eq(3*x(t) + 8*y(t) + 2*Derivative(x(t), t) + 2*Derivative(y(t), t) - 2,0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - \frac {4 C_{1} \left (1 + \sqrt {10}\right ) e^{- \frac {t \left (2 - 2 \sqrt {10}\right )}{3}}}{9} - \frac {4 C_{2} \left (1 - \sqrt {10}\right ) e^{- \frac {t \left (2 + 2 \sqrt {10}\right )}{3}}}{9} - \frac {t}{3} - \frac {7}{36}, \ y{\left (t \right )} = C_{1} e^{- \frac {t \left (2 - 2 \sqrt {10}\right )}{3}} + C_{2} e^{- \frac {t \left (2 + 2 \sqrt {10}\right )}{3}} + \frac {t}{8} + \frac {3}{8}\right ] \]