60.9.29 problem 1884

Internal problem ID [11808]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 8, system of first order odes
Problem number : 1884
Date solved : Sunday, March 30, 2025 at 09:15:48 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.269 (sec). Leaf size: 68

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

\begin{align*} x \left (t \right ) &= -t^{2}+2 \,{\mathrm e}^{\frac {t}{2}} c_1 +\frac {2 \cos \left (2 t \right )}{17}+\frac {\sin \left (2 t \right )}{34}-4 t +c_2 \\ y \left (t \right ) &= -t +\frac {{\mathrm e}^{\frac {t}{2}} c_1}{2}+\frac {9 \sin \left (2 t \right )}{68}+\frac {\cos \left (2 t \right )}{34}+2-\frac {t^{2}}{2}+\frac {c_2}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.426 (sec). Leaf size: 199

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

\begin{align*} x(t)\to 8 \int _1^t-\frac {1}{8} e^{-\frac {K[1]}{2}} \left (e^{t/2}-8 e^{\frac {K[1]}{2}}\right ) (\cos (2 K[1])-2 K[1])dK[1]+7 \int _1^t(2 K[2]-\cos (2 K[2]))dK[2]+8 c_1 e^{t/2}+8 c_2 e^{t/2}-c_2 \\ y(t)\to 2 \left (\int _1^t-\frac {1}{8} e^{-\frac {K[1]}{2}} \left (e^{t/2}-8 e^{\frac {K[1]}{2}}\right ) (\cos (2 K[1])-2 K[1])dK[1]+(c_1+c_2) e^{t/2}-c_2\right )+\frac {3}{2} \left (\int _1^t(2 K[2]-\cos (2 K[2]))dK[2]+c_2\right ) \\ \end{align*}

✗ Sympy

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

KeyError : Derivative(y(t), t)