60.9.41 problem 1896

Internal problem ID [11820]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 8, system of first order odes
Problem number : 1896
Date solved : Friday, March 14, 2025 at 02:58:30 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.072 (sec). Leaf size: 74

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

\begin{align*} x \left (t \right ) &= -2-t -\frac {2 c_{2} {\mathrm e}^{-t}}{3}-c_3 \,{\mathrm e}^{t}-2 c_4 t \,{\mathrm e}^{t}-3 c_5 \,{\mathrm e}^{t} t^{2}-6 c_5 \,{\mathrm e}^{t} \\ y \left (t \right ) &= -2+c_{1} {\mathrm e}^{t}+c_{2} {\mathrm e}^{-t}+c_3 \,{\mathrm e}^{t} t +c_4 \,t^{2} {\mathrm e}^{t}+c_5 \,{\mathrm e}^{t} t^{3} \\ \end{align*}

✓ Mathematica. Time used: 0.652 (sec). Leaf size: 892

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

\begin{align*} \text {Solution too large to show}\end{align*}

✓ Sympy. Time used: 0.459 (sec). Leaf size: 78

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

\[ \left [ x{\left (t \right )} = C_{1} t e^{t} + \frac {C_{2} t^{2} e^{t}}{2} - \frac {2 C_{3} e^{- t}}{3} - t + \left (C_{2} + C_{4}\right ) e^{t} - 2, \ y{\left (t \right )} = - \frac {C_{1} t^{2} e^{t}}{2} - \frac {C_{2} t^{3} e^{t}}{6} + C_{3} e^{- t} - C_{4} t e^{t} - C_{5} e^{t} - 2\right ] \]