67.6.5 problem Problem 4(e)

Internal problem ID [14030]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 6.4 Reduction to a single ODE. Problems page 415
Problem number : Problem 4(e)
Date solved : Wednesday, March 05, 2025 at 10:26:31 PM
CAS classification : system_of_ODEs

\begin{align*} 5 \frac {d}{d t}x \left (t \right )-3 \frac {d}{d t}y \left (t \right )&=x \left (t \right )+y \left (t \right )\\ 3 \frac {d}{d t}x \left (t \right )-\frac {d}{d t}y \left (t \right )&=t \end{align*}

✓ Maple. Time used: 0.055 (sec). Leaf size: 44

ode:=[5*diff(x(t),t)-3*diff(y(t),t) = x(t)+y(t), 3*diff(x(t),t)-diff(y(t),t) = t]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= \frac {t^{2}}{8}-{\mathrm e}^{-t} c_{1} +\frac {t}{2}+c_{2} \\ y &= \frac {3 t}{2}-3 \,{\mathrm e}^{-t} c_{1} -2-\frac {t^{2}}{8}-c_{2} \\ \end{align*}

✓ Mathematica. Time used: 0.084 (sec). Leaf size: 179

ode={5*D[x[t],t]-3*D[y[t],t]==x[t]+y[t],3*D[x[t],t]-D[y[t],t]==t}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to \frac {1}{4} e^{-t} \left (\left (3 e^t+1\right ) \int _1^t\frac {1}{4} \left (1+2 e^{K[1]}\right ) K[1]dK[1]-\left (e^t-1\right ) \int _1^t\frac {1}{4} \left (-1+6 e^{K[2]}\right ) K[2]dK[2]+3 c_1 e^t-c_2 e^t+c_1+c_2\right ) \\ y(t)\to \frac {1}{4} e^{-t} \left (-3 \left (e^t-1\right ) \int _1^t\frac {1}{4} \left (1+2 e^{K[1]}\right ) K[1]dK[1]+\left (e^t+3\right ) \int _1^t\frac {1}{4} \left (-1+6 e^{K[2]}\right ) K[2]dK[2]-3 c_1 e^t+c_2 e^t+3 c_1+3 c_2\right ) \\ \end{align*}

✓ Sympy. Time used: 0.164 (sec). Leaf size: 44

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

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