61.6.4 problem Problem 4(d)

Internal problem ID [15389]
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(d)
Date solved : Thursday, October 02, 2025 at 10:12:29 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.121 (sec). Leaf size: 36

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

\begin{align*} x \left (t \right ) &= \frac {t^{2}}{4}+\frac {7 \,{\mathrm e}^{\frac {2 t}{7}} c_1}{2}+\frac {3 t}{4}+c_2 \\ y \left (t \right ) &= \frac {3 t}{2}+7 \,{\mathrm e}^{\frac {2 t}{7}} c_1 +\frac {21}{4} \\ \end{align*}

✓ Mathematica. Time used: 0.084 (sec). Leaf size: 117

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

\begin{align*} x(t)&\to \int _1^t\frac {1}{14} \left (7-3 e^{-\frac {2 K[1]}{7}}\right ) K[1]dK[1]+\frac {1}{2} \left (e^{2 t/7}-1\right ) \int _1^t-\frac {3}{7} e^{-\frac {2 K[2]}{7}} K[2]dK[2]+\frac {1}{2} c_2 \left (e^{2 t/7}-1\right )+c_1\\ y(t)&\to e^{2 t/7} \left (\int _1^t-\frac {3}{7} e^{-\frac {2 K[2]}{7}} K[2]dK[2]+c_2\right ) \end{align*}

✓ Sympy. Time used: 0.081 (sec). Leaf size: 46

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

\[ \left [ x{\left (t \right )} = C_{1} + \frac {C_{2} e^{\frac {2 t}{7}}}{2} + \frac {t^{2}}{4} + \frac {3 t}{4} + \frac {21}{8}, \ y{\left (t \right )} = C_{2} e^{\frac {2 t}{7}} + \frac {3 t}{2} + \frac {21}{4}\right ] \]