76.56.1 problem Ex. 1

Internal problem ID [20326]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter XI. Ordinary differential equations with more than two variables. End of chapter problems at page 143
Problem number : Ex. 1
Date solved : Thursday, October 02, 2025 at 05:41:39 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.211 (sec). Leaf size: 42

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

\begin{align*} x \left (t \right ) &= \frac {{\mathrm e}^{2 t}}{2}-\frac {3 \,{\mathrm e}^{t}}{11}+{\mathrm e}^{-\frac {6 t}{5}} c_2 \\ y \left (t \right ) &= -8 \,{\mathrm e}^{-\frac {6 t}{5}} c_2 +\frac {15 \,{\mathrm e}^{t}}{22}+c_1 \,{\mathrm e}^{-t} \\ \end{align*}

✓ Mathematica. Time used: 0.3 (sec). Leaf size: 65

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

\begin{align*} x(t)&\to \frac {1}{22} e^t \left (11 e^t-6\right )+c_1 e^{-6 t/5}\\ y(t)&\to \frac {15 e^t}{22}-8 c_1 e^{-6 t/5}+(8 c_1+c_2) e^{-t} \end{align*}

✓ Sympy. Time used: 0.151 (sec). Leaf size: 48

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

\[ \left [ x{\left (t \right )} = - \frac {C_{1} e^{- \frac {6 t}{5}}}{8} + \frac {e^{2 t}}{2} - \frac {3 e^{t}}{11}, \ y{\left (t \right )} = C_{1} e^{- \frac {6 t}{5}} + C_{2} e^{- t} + \frac {15 e^{t}}{22}\right ] \]