67.6.3 problem Problem 4(c)

Internal problem ID [14036]
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(c)
Date solved : Monday, March 31, 2025 at 08:22:36 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 )+y \left (t \right )-t\\ 2 \frac {d}{d t}x \left (t \right )+3 \frac {d}{d t}y \left (t \right )&=2 x \left (t \right )+6 \end{align*}

✓ Maple. Time used: 0.125 (sec). Leaf size: 29

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

\begin{align*} x \left (t \right ) &= {\mathrm e}^{t} c_2 +{\mathrm e}^{-\frac {2 t}{5}} c_1 -\frac {3}{2} \\ y \left (t \right ) &= -\frac {7 \,{\mathrm e}^{-\frac {2 t}{5}} c_1}{3}+\frac {1}{2}+t \\ \end{align*}

✓ Mathematica. Time used: 0.352 (sec). Leaf size: 152

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

\begin{align*} x(t)\to e^t \int _1^t-\frac {3}{35} e^{-K[1]} \left (5 (K[1]-4)+2 e^{\frac {7 K[1]}{5}} (K[1]+3)\right )dK[1]+\frac {1}{7} e^{-2 t/5} \left (3 \left (e^{7 t/5}-1\right ) \int _1^t\frac {2}{5} e^{\frac {2 K[2]}{5}} (K[2]+3)dK[2]+(7 c_1+3 c_2) e^{7 t/5}-3 c_2\right ) \\ y(t)\to e^{-2 t/5} \left (\int _1^t\frac {2}{5} e^{\frac {2 K[2]}{5}} (K[2]+3)dK[2]+c_2\right ) \\ \end{align*}

✓ Sympy. Time used: 0.205 (sec). Leaf size: 37

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

\[ \left [ x{\left (t \right )} = - \frac {3 C_{1} e^{- \frac {2 t}{5}}}{7} + C_{2} e^{t} - \frac {3}{2}, \ y{\left (t \right )} = C_{1} e^{- \frac {2 t}{5}} + t + \frac {1}{2}\right ] \]