60.9.28 problem 1883

Internal problem ID [11807]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 8, system of first order odes
Problem number : 1883
Date solved : Sunday, March 30, 2025 at 09:15:47 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )-\frac {d}{d t}y \left (t \right )+x \left (t \right )&=2 t\\ \frac {d^{2}}{d t^{2}}x \left (t \right )+\frac {d}{d t}y \left (t \right )-9 x \left (t \right )+3 y \left (t \right )&=\sin \left (2 t \right ) \end{align*}

✓ Maple. Time used: 0.299 (sec). Leaf size: 79

ode:=[diff(x(t),t)-diff(y(t),t)+x(t) = 2*t, diff(diff(x(t),t),t)+diff(y(t),t)-9*x(t)+3*y(t) = sin(2*t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= 2 t +4-\frac {36 \sin \left (2 t \right )}{325}-\frac {2 \cos \left (2 t \right )}{325}+c_1 \,{\mathrm e}^{t}+c_2 \,{\mathrm e}^{-3 t}+c_3 \,{\mathrm e}^{t} t \\ y \left (t \right ) &= -\frac {37 \sin \left (2 t \right )}{325}+\frac {16 \cos \left (2 t \right )}{325}+2 c_1 \,{\mathrm e}^{t}+\frac {2 c_2 \,{\mathrm e}^{-3 t}}{3}+2 c_3 \,{\mathrm e}^{t} t -c_3 \,{\mathrm e}^{t}+10+6 t \\ \end{align*}

✓ Mathematica. Time used: 0.981 (sec). Leaf size: 488

ode={D[x[t],t]-D[y[t],t]+x[t]==2*t,D[x[t],{t,2}]+D[y[t],t]-9*x[t]+3*y[t]==Sin[2*t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to \frac {1}{16} e^{-3 t} \left (\left (e^{4 t} (20 t+7)+9\right ) \int _1^t\frac {1}{16} e^{-K[1]} \left (\left (-4 K[1]-3 e^{4 K[1]}+3\right ) \sin (2 K[1])-32 K[1]^2\right )dK[1]+\left (e^{4 t} (4 t+3)-3\right ) \int _1^t\frac {1}{16} e^{-K[2]} \left (\left (-4 K[2]+9 e^{4 K[2]}+7\right ) \sin (2 K[2])-32 (K[2]-1) K[2]\right )dK[2]-3 \left (e^{4 t} (4 t-1)+1\right ) \int _1^t\frac {1}{8} e^{-K[3]} \left (-16 K[3] (2 K[3]+1)-\left (4 K[3]+e^{4 K[3]}-1\right ) \sin (2 K[3])\right )dK[3]+c_1 \left (e^{4 t} (20 t+7)+9\right )+c_2 \left (e^{4 t} (4 t+3)-3\right )-3 c_3 \left (e^{4 t} (4 t-1)+1\right )\right ) \\ y(t)\to \frac {1}{8} e^{-3 t} \left (\left (e^{4 t} (20 t-3)+3\right ) \int _1^t\frac {1}{16} e^{-K[1]} \left (\left (-4 K[1]-3 e^{4 K[1]}+3\right ) \sin (2 K[1])-32 K[1]^2\right )dK[1]+\left (e^{4 t} (4 t+1)-1\right ) \int _1^t\frac {1}{16} e^{-K[2]} \left (\left (-4 K[2]+9 e^{4 K[2]}+7\right ) \sin (2 K[2])-32 (K[2]-1) K[2]\right )dK[2]-\left (3 e^{4 t} (4 t-3)+1\right ) \int _1^t\frac {1}{8} e^{-K[3]} \left (-16 K[3] (2 K[3]+1)-\left (4 K[3]+e^{4 K[3]}-1\right ) \sin (2 K[3])\right )dK[3]+c_1 \left (e^{4 t} (20 t-3)+3\right )+c_2 \left (e^{4 t} (4 t+1)-1\right )-c_3 \left (3 e^{4 t} (4 t-3)+1\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.622 (sec). Leaf size: 105

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

\[ \left [ x{\left (t \right )} = \frac {2 C_{1} t e^{t}}{3} + \frac {3 C_{2} e^{- 3 t}}{2} + 2 t + \left (\frac {C_{1}}{3} + \frac {2 C_{3}}{3}\right ) e^{t} - \frac {36 \sin {\left (2 t \right )}}{325} - \frac {2 \cos {\left (2 t \right )}}{325} + 4, \ y{\left (t \right )} = \frac {4 C_{1} t e^{t}}{3} + C_{2} e^{- 3 t} + \frac {4 C_{3} e^{t}}{3} + 6 t - \frac {37 \sin {\left (2 t \right )}}{325} + \frac {16 \cos {\left (2 t \right )}}{325} + 10\right ] \]