15.17.5 problem 5

Internal problem ID [3241]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 26, page 115
Problem number : 5
Date solved : Sunday, March 30, 2025 at 01:23:38 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.182 (sec). Leaf size: 49

ode:=[2*diff(x(t),t)+3*x(t)-y(t) = exp(t), 5*x(t)-3*diff(y(t),t) = y(t)+2*t]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= \frac {3 \,{\mathrm e}^{\frac {t}{6}} c_2}{10}-{\mathrm e}^{-2 t} c_1 +\frac {11}{2}+\frac {4 \,{\mathrm e}^{t}}{15}+t \\ y \left (t \right ) &= {\mathrm e}^{\frac {t}{6}} c_2 +{\mathrm e}^{-2 t} c_1 +3 t +\frac {37}{2}+\frac {{\mathrm e}^{t}}{3} \\ \end{align*}

✓ Mathematica. Time used: 0.788 (sec). Leaf size: 105

ode={2*D[x[t],t]+3*x[t]-y[t]==Exp[t],5*x[t]-3*D[y[t],t]==y[t]+2*t}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to t+\frac {4 e^t}{15}+\frac {1}{13} (10 c_1-3 c_2) e^{-2 t}+\frac {3}{13} (c_1+c_2) e^{t/6}+\frac {11}{2} \\ y(t)\to \frac {1}{78} e^{-2 t} \left (39 e^{2 t} (6 t+37)+26 e^{3 t}+60 (c_1+c_2) e^{13 t/6}-60 c_1+18 c_2\right ) \\ \end{align*}

✓ Sympy. Time used: 0.274 (sec). Leaf size: 58

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(3*x(t) - y(t) - exp(t) + 2*Derivative(x(t), t),0),Eq(-2*t + 5*x(t) - y(t) - 3*Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - C_{1} e^{- 2 t} + \frac {3 C_{2} e^{\frac {t}{6}}}{10} + t + \frac {4 e^{t}}{15} + \frac {11}{2}, \ y{\left (t \right )} = C_{1} e^{- 2 t} + C_{2} e^{\frac {t}{6}} + 3 t + \frac {e^{t}}{3} + \frac {37}{2}\right ] \]