56.39.1 problem Ex 1

Internal problem ID [14305]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter X, System of simulataneous equations. Article 64. Systems of linear equations with constant coefficients. Page 150
Problem number : Ex 1
Date solved : Thursday, October 02, 2025 at 09:30:27 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.183 (sec). Leaf size: 46

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

\begin{align*} x \left (t \right ) &= -\frac {{\mathrm e}^{\frac {t}{3}} c_2}{2}-{\mathrm e}^{-\frac {t}{3}} c_1 -6 t \\ y \left (t \right ) &= {\mathrm e}^{\frac {t}{3}} c_2 +{\mathrm e}^{-\frac {t}{3}} c_1 +9 t +9+\frac {{\mathrm e}^{t}}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.95 (sec). Leaf size: 90

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

\begin{align*} x(t)&\to e^{-t/3} \left (-6 e^{t/3} t-(c_1+c_2) e^{2 t/3}+2 c_1+c_2\right )\\ y(t)&\to 9 (t+1)+\frac {e^t}{2}+2 (c_1+c_2) e^{t/3}-(2 c_1+c_2) e^{-t/3} \end{align*}

✓ Sympy. Time used: 0.143 (sec). Leaf size: 48

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

\[ \left [ x{\left (t \right )} = - C_{1} e^{- \frac {t}{3}} - \frac {C_{2} e^{\frac {t}{3}}}{2} - 6 t, \ y{\left (t \right )} = C_{1} e^{- \frac {t}{3}} + C_{2} e^{\frac {t}{3}} + 9 t + \frac {e^{t}}{2} + 9\right ] \]