60.9.36 problem 1891

Internal problem ID [11815]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 8, system of first order odes
Problem number : 1891
Date solved : Sunday, March 30, 2025 at 09:15:55 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d^{2}}{d t^{2}}x \left (t \right )+6 x \left (t \right )+7 y \left (t \right )&=0\\ \frac {d^{2}}{d t^{2}}y \left (t \right )+3 x \left (t \right )+2 y \left (t \right )&=2 t \end{align*}

✓ Maple. Time used: 0.169 (sec). Leaf size: 63

ode:=[diff(diff(x(t),t),t)+6*x(t)+7*y(t) = 0, diff(diff(y(t),t),t)+3*x(t)+2*y(t) = 2*t]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= \frac {14 t}{9}+c_1 \,{\mathrm e}^{t}+c_2 \cos \left (3 t \right )+c_3 \,{\mathrm e}^{-t}+c_4 \sin \left (3 t \right ) \\ y \left (t \right ) &= -c_1 \,{\mathrm e}^{t}+\frac {3 c_2 \cos \left (3 t \right )}{7}-c_3 \,{\mathrm e}^{-t}+\frac {3 c_4 \sin \left (3 t \right )}{7}-\frac {4 t}{3} \\ \end{align*}

✓ Mathematica. Time used: 0.695 (sec). Leaf size: 664

ode={D[x[t],{t,2}]+6*x[t]+7*y[t]==0,D[y[t],{t,2}]+3*x[t]+2*y[t]==2*t}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to \frac {1}{60} e^{-t} \left (-21 \left (e^{2 t}-2 e^t \cos (3 t)+1\right ) \int _1^t-\frac {1}{10} e^{-K[3]} K[3] \left (2 e^{K[3]} \sin (3 K[3])+7 e^{2 K[3]}-7\right )dK[3]+\left (9 e^{2 t}+14 e^t \sin (3 t)-9\right ) \int _1^t-\frac {7}{10} e^{-K[2]} \left (-2 e^{K[2]} \cos (3 K[2])+e^{2 K[2]}+1\right ) K[2]dK[2]+7 \left (-3 e^{2 t}+2 e^t \sin (3 t)+3\right ) \int _1^t\frac {1}{10} e^{-K[4]} \left (6 e^{K[4]} \cos (3 K[4])+7 e^{2 K[4]}+7\right ) K[4]dK[4]+3 \left (3 e^{2 t}+14 e^t \cos (3 t)+3\right ) \int _1^t\frac {7}{30} e^{-K[1]} K[1] \left (-2 e^{K[1]} \sin (3 K[1])+3 e^{2 K[1]}-3\right )dK[1]-21 c_3 \left (e^{2 t}-2 e^t \cos (3 t)+1\right )+3 c_1 \left (3 e^{2 t}+14 e^t \cos (3 t)+3\right )+7 c_4 \left (-3 e^{2 t}+2 e^t \sin (3 t)+3\right )+c_2 \left (9 e^{2 t}+14 e^t \sin (3 t)-9\right )\right ) \\ y(t)\to \frac {1}{20} e^{-t} \left (-3 \left (e^{2 t}-2 e^t \cos (3 t)+1\right ) \int _1^t\frac {7}{30} e^{-K[1]} K[1] \left (-2 e^{K[1]} \sin (3 K[1])+3 e^{2 K[1]}-3\right )dK[1]+\left (-3 e^{2 t}+2 e^t \sin (3 t)+3\right ) \int _1^t-\frac {7}{10} e^{-K[2]} \left (-2 e^{K[2]} \cos (3 K[2])+e^{2 K[2]}+1\right ) K[2]dK[2]+\left (7 e^{2 t}+2 e^t \sin (3 t)-7\right ) \int _1^t\frac {1}{10} e^{-K[4]} \left (6 e^{K[4]} \cos (3 K[4])+7 e^{2 K[4]}+7\right ) K[4]dK[4]+\left (7 e^{2 t}+6 e^t \cos (3 t)+7\right ) \int _1^t-\frac {1}{10} e^{-K[3]} K[3] \left (2 e^{K[3]} \sin (3 K[3])+7 e^{2 K[3]}-7\right )dK[3]-3 c_1 \left (e^{2 t}-2 e^t \cos (3 t)+1\right )+c_3 \left (7 e^{2 t}+6 e^t \cos (3 t)+7\right )+c_2 \left (-3 e^{2 t}+2 e^t \sin (3 t)+3\right )+c_4 \left (7 e^{2 t}+2 e^t \sin (3 t)-7\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.311 (sec). Leaf size: 116

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(6*x(t) + 7*y(t) + Derivative(x(t), (t, 2)),0),Eq(-2*t + 3*x(t) + 2*y(t) + Derivative(y(t), (t, 2)),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = C_{1} e^{- t} - C_{2} e^{t} + \frac {7 C_{3} \sin {\left (3 t \right )}}{9} + \frac {7 C_{4} \cos {\left (3 t \right )}}{9} + \frac {7 t \sin ^{2}{\left (3 t \right )}}{45} + \frac {7 t \cos ^{2}{\left (3 t \right )}}{45} + \frac {7 t}{5}, \ y{\left (t \right )} = - C_{1} e^{- t} + C_{2} e^{t} + \frac {C_{3} \sin {\left (3 t \right )}}{3} + \frac {C_{4} \cos {\left (3 t \right )}}{3} + \frac {t \sin ^{2}{\left (3 t \right )}}{15} + \frac {t \cos ^{2}{\left (3 t \right )}}{15} - \frac {7 t}{5}\right ] \]