67.6.6 problem Problem 4(f)

Internal problem ID [14039]
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(f)
Date solved : Monday, March 31, 2025 at 08:22:40 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )-4 \frac {d}{d t}y \left (t \right )&=0\\ 2 \frac {d}{d t}x \left (t \right )-3 \frac {d}{d t}y \left (t \right )&=y \left (t \right )+t \end{align*}

✓ Maple. Time used: 0.128 (sec). Leaf size: 31

ode:=[diff(x(t),t)-4*diff(y(t),t) = 0, 2*diff(x(t),t)-3*diff(y(t),t) = y(t)+t]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= 5 \,{\mathrm e}^{\frac {t}{5}} c_1 -4 t +c_2 \\ y \left (t \right ) &= \frac {5 \,{\mathrm e}^{\frac {t}{5}} c_1}{4}-5-t \\ \end{align*}

✓ Mathematica. Time used: 0.023 (sec). Leaf size: 109

ode={D[x[t],t]-4*D[y[t],t]==0,2*D[x[t],t]-3*D[y[t],t]==y[t]+t}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.137 (sec). Leaf size: 29

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(Derivative(x(t), t) - 4*Derivative(y(t), t),0),Eq(-t - y(t) + 2*Derivative(x(t), t) - 3*Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = C_{1} + 4 C_{2} e^{\frac {t}{5}} - 4 t - 20, \ y{\left (t \right )} = C_{2} e^{\frac {t}{5}} - t - 5\right ] \]