60.9.18 problem 1873

Internal problem ID [11797]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 8, system of first order odes
Problem number : 1873
Date solved : Wednesday, March 05, 2025 at 03:07:10 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: 0.081 (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 ) &= c_{2} {\mathrm e}^{-t}+{\mathrm e}^{-6 t} c_{1} -\frac {29 \,{\mathrm e}^{t}}{7}+\frac {19 t}{3}-\frac {56}{9} \\ y \left (t \right ) &= -c_{2} {\mathrm e}^{-t}+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: 0.361 (sec). Leaf size: 291

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}{5} e^{-6 t} \left (\left (4 e^{5 t}+1\right ) \int _1^t\frac {1}{5} e^{K[1]} \left (4 e^{5 K[1]} K[1]+31 K[1]-40 e^{K[1]}-5 e^{6 K[1]}\right )dK[1]-\left (e^{5 t}-1\right ) \int _1^t\left (-\frac {31}{5} e^{K[2]} K[2]+\frac {16}{5} e^{6 K[2]} K[2]+8 e^{2 K[2]}-4 e^{7 K[2]}\right )dK[2]+4 c_1 e^{5 t}-c_2 e^{5 t}+c_1+c_2\right ) \\ y(t)\to \frac {1}{5} e^{-6 t} \left (-4 \left (e^{5 t}-1\right ) \int _1^t\frac {1}{5} e^{K[1]} \left (4 e^{5 K[1]} K[1]+31 K[1]-40 e^{K[1]}-5 e^{6 K[1]}\right )dK[1]+\left (e^{5 t}+4\right ) \int _1^t\left (-\frac {31}{5} e^{K[2]} K[2]+\frac {16}{5} e^{6 K[2]} K[2]+8 e^{2 K[2]}-4 e^{7 K[2]}\right )dK[2]-4 c_1 e^{5 t}+c_2 e^{5 t}+4 c_1+4 c_2\right ) \\ \end{align*}

✓ Sympy. Time used: 0.265 (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)
 

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