60.9.17 problem 1872

Internal problem ID [11796]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 8, system of first order odes
Problem number : 1872
Date solved : Wednesday, March 05, 2025 at 03:07:09 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 )+11 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 )+8 x \left (t \right )+24 y \left (t \right )&={\mathrm e}^{2 t} \end{align*}

✓ Maple. Time used: 0.083 (sec). Leaf size: 64

ode:=[4*diff(x(t),t)+9*diff(y(t),t)+11*x(t)+31*y(t) = exp(t), 3*diff(x(t),t)+7*diff(y(t),t)+8*x(t)+24*y(t) = exp(2*t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= c_{2} {\mathrm e}^{-4 t}+{\mathrm e}^{-4 t} t c_{1} +\frac {31 \,{\mathrm e}^{t}}{25}-\frac {49 \,{\mathrm e}^{2 t}}{36} \\ y \left (t \right ) &= \frac {19 \,{\mathrm e}^{2 t}}{36}-c_{2} {\mathrm e}^{-4 t}-{\mathrm e}^{-4 t} t c_{1} -\frac {11 \,{\mathrm e}^{t}}{25}-{\mathrm e}^{-4 t} c_{1} \\ \end{align*}

✓ Mathematica. Time used: 0.23 (sec). Leaf size: 184

ode={4*D[x[t],t]+9*D[y[t],t]+11*x[t]+31*y[t]==Exp[t],3*D[x[t],t]+7*D[y[t],t]+8*x[t]+24*y[t]==Exp[2*t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to e^{-4 t} \left (-(t-1) \int _1^te^{5 K[1]} \left (4 K[1]-e^{K[1]} (5 K[1]+9)+7\right )dK[1]-t \int _1^te^{5 K[2]} \left (-4 K[2]+e^{K[2]} (5 K[2]+4)-3\right )dK[2]+c_1 (-t)-c_2 t+c_1\right ) \\ y(t)\to e^{-4 t} \left (t \int _1^te^{5 K[1]} \left (4 K[1]-e^{K[1]} (5 K[1]+9)+7\right )dK[1]+(t+1) \int _1^te^{5 K[2]} \left (-4 K[2]+e^{K[2]} (5 K[2]+4)-3\right )dK[2]+c_1 t+c_2 t+c_2\right ) \\ \end{align*}

✓ Sympy. Time used: 0.257 (sec). Leaf size: 66

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(11*x(t) + 31*y(t) - exp(t) + 4*Derivative(x(t), t) + 9*Derivative(y(t), t),0),Eq(8*x(t) + 24*y(t) - exp(2*t) + 3*Derivative(x(t), t) + 7*Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - C_{2} t e^{- 4 t} - \left (C_{1} - C_{2}\right ) e^{- 4 t} - \frac {49 e^{2 t}}{36} + \frac {31 e^{t}}{25}, \ y{\left (t \right )} = C_{1} e^{- 4 t} + C_{2} t e^{- 4 t} + \frac {19 e^{2 t}}{36} - \frac {11 e^{t}}{25}\right ] \]