82.55.3 problem Ex. 3

Internal problem ID [18967]
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. problems at page 129
Problem number : Ex. 3
Date solved : Monday, March 31, 2025 at 06:26:51 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.177 (sec). Leaf size: 50

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

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-5 t} c_2 +{\mathrm e}^{t} c_1 -\frac {13}{25}+\frac {3 \,{\mathrm e}^{2 t}}{7}-\frac {2 t}{5} \\ y \left (t \right ) &= -{\mathrm e}^{-5 t} c_2 +{\mathrm e}^{t} c_1 +\frac {4 \,{\mathrm e}^{2 t}}{7}-\frac {12}{25}-\frac {3 t}{5} \\ \end{align*}

✓ Mathematica. Time used: 0.74 (sec). Leaf size: 101

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

\begin{align*} x(t)\to -\frac {2 t}{5}+\frac {3 e^{2 t}}{7}+\frac {1}{2} (c_1-c_2) e^{-5 t}+\frac {1}{2} (c_1+c_2) e^t-\frac {13}{25} \\ y(t)\to -\frac {3}{25} (5 t+4)+\frac {4 e^{2 t}}{7}+\frac {1}{2} (c_2-c_1) e^{-5 t}+\frac {1}{2} (c_1+c_2) e^t \\ \end{align*}

✓ Sympy. Time used: 0.227 (sec). Leaf size: 61

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

\[ \left [ x{\left (t \right )} = - C_{1} e^{- 5 t} + C_{2} e^{t} - \frac {2 t}{5} + \frac {3 e^{2 t}}{7} - \frac {13}{25}, \ y{\left (t \right )} = C_{1} e^{- 5 t} + C_{2} e^{t} - \frac {3 t}{5} + \frac {4 e^{2 t}}{7} - \frac {12}{25}\right ] \]