82.55.4 problem Ex. 4

Internal problem ID [18968]
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. 4
Date solved : Monday, March 31, 2025 at 06:26:52 PM
CAS classification : system_of_ODEs

\begin{align*} 4 \frac {d}{d t}x \left (t \right )+9 \frac {d}{d t}y \left (t \right )+44 x \left (t \right )+49 y \left (t \right )&=t\\ 3 \frac {d}{d t}x \left (t \right )+7 \frac {d}{d t}y \left (t \right )+34 x \left (t \right )+38 y \left (t \right )&={\mathrm e}^{t} \end{align*}

✓ Maple. Time used: 0.179 (sec). Leaf size: 51

ode:=[4*diff(x(t),t)+9*diff(y(t),t)+44*x(t)+49*y(t) = t, 3*diff(x(t),t)+7*diff(y(t),t)+34*x(t)+38*y(t) = exp(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-t} c_2 +{\mathrm e}^{-6 t} c_1 -\frac {29 \,{\mathrm e}^{t}}{7}+\frac {19 t}{3}-\frac {56}{9} \\ y \left (t \right ) &= -{\mathrm e}^{-t} c_2 +4 \,{\mathrm e}^{-6 t} c_1 +\frac {24 \,{\mathrm e}^{t}}{7}+\frac {55}{9}-\frac {17 t}{3} \\ \end{align*}

✓ Mathematica. Time used: 0.162 (sec). Leaf size: 104

ode={4*D[x[t],t]+9*D[y[t],t]+44*x[t]+49*y[t]==t,3*D[x[t],t]+7*D[y[t],t]+34*x[t]+38*y[t]==Exp[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to \frac {1}{9} (57 t-56)-\frac {29 e^t}{7}+\frac {1}{5} (4 c_1-c_2) e^{-t}+\frac {1}{5} (c_1+c_2) e^{-6 t} \\ y(t)\to \frac {1}{9} (55-51 t)+\frac {24 e^t}{7}+\frac {1}{5} (c_2-4 c_1) e^{-t}+\frac {4}{5} (c_1+c_2) e^{-6 t} \\ \end{align*}

✓ Sympy. Time used: 0.239 (sec). Leaf size: 60

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-t + 44*x(t) + 49*y(t) + 4*Derivative(x(t), t) + 9*Derivative(y(t), t),0),Eq(34*x(t) + 38*y(t) - exp(t) + 3*Derivative(x(t), t) + 7*Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - C_{1} e^{- t} + \frac {C_{2} e^{- 6 t}}{4} + \frac {19 t}{3} - \frac {29 e^{t}}{7} - \frac {56}{9}, \ y{\left (t \right )} = C_{1} e^{- t} + C_{2} e^{- 6 t} - \frac {17 t}{3} + \frac {24 e^{t}}{7} + \frac {55}{9}\right ] \]