76.56.2 problem Ex. 2

Internal problem ID [20327]
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. 2
Date solved : Thursday, October 02, 2025 at 05:41:40 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 )+2 x \left (t \right )+31 y \left (t \right )&={\mathrm e}^{t}\\ 3 \frac {d}{d t}x \left (t \right )+7 \frac {d}{d t}y \left (t \right )+x \left (t \right )+24 y \left (t \right )&=3 \end{align*}

✓ Maple. Time used: 0.273 (sec). Leaf size: 70

ode:=[4*diff(x(t),t)+9*diff(y(t),t)+2*x(t)+31*y(t) = exp(t), 3*diff(x(t),t)+7*diff(y(t),t)+x(t)+24*y(t) = 3]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-4 t} \sin \left (t \right ) c_2 +{\mathrm e}^{-4 t} \cos \left (t \right ) c_1 -\frac {93}{17}+\frac {31 \,{\mathrm e}^{t}}{26} \\ y \left (t \right ) &= -{\mathrm e}^{-4 t} \sin \left (t \right ) c_2 -{\mathrm e}^{-4 t} \cos \left (t \right ) c_2 -{\mathrm e}^{-4 t} \cos \left (t \right ) c_1 +{\mathrm e}^{-4 t} \sin \left (t \right ) c_1 -\frac {2 \,{\mathrm e}^{t}}{13}+\frac {6}{17} \\ \end{align*}

✓ Mathematica. Time used: 0.137 (sec). Leaf size: 79

ode={4*D[x[t],t]+9*D[y[t],t]+2*x[t]+31*y[t]==Exp[t],3*D[x[t],t]+7*D[y[t],t]+x[t]+24*y[t]==3}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {31 e^t}{26}+c_1 e^{-4 t} \cos (t)-(c_1+c_2) e^{-4 t} \sin (t)-\frac {93}{17}\\ y(t)&\to -\frac {2 e^t}{13}+c_2 e^{-4 t} \cos (t)+(2 c_1+c_2) e^{-4 t} \sin (t)+\frac {6}{17} \end{align*}

✓ Sympy. Time used: 2.872 (sec). Leaf size: 136

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(2*x(t) + 31*y(t) - exp(t) + 4*Derivative(x(t), t) + 9*Derivative(y(t), t),0),Eq(x(t) + 24*y(t) + 3*Derivative(x(t), t) + 7*Derivative(y(t), t) - 3,0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - \left (\frac {C_{1}}{2} - \frac {C_{2}}{2}\right ) e^{- 4 t} \sin {\left (t \right )} - \left (\frac {C_{1}}{2} + \frac {C_{2}}{2}\right ) e^{- 4 t} \cos {\left (t \right )} + \frac {31 e^{t} \sin ^{2}{\left (t \right )}}{26} + \frac {31 e^{t} \cos ^{2}{\left (t \right )}}{26} - \frac {93 \sin ^{2}{\left (t \right )}}{17} - \frac {93 \cos ^{2}{\left (t \right )}}{17}, \ y{\left (t \right )} = C_{1} e^{- 4 t} \cos {\left (t \right )} - C_{2} e^{- 4 t} \sin {\left (t \right )} - \frac {2 e^{t} \sin ^{2}{\left (t \right )}}{13} - \frac {2 e^{t} \cos ^{2}{\left (t \right )}}{13} + \frac {6 \sin ^{2}{\left (t \right )}}{17} + \frac {6 \cos ^{2}{\left (t \right )}}{17}\right ] \]