60.9.15 problem 1870

Internal problem ID [11794]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 8, system of first order odes
Problem number : 1870
Date solved : Wednesday, March 05, 2025 at 03:07:07 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.276 (sec). Leaf size: 46

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

\begin{align*} x \left (t \right ) &= \frac {{\mathrm e}^{4 t} c_{1}}{4}+\frac {5 \sin \left (t \right )}{17}+{\mathrm e}^{t}-\frac {3 \cos \left (t \right )}{17}+c_{2} \\ y \left (t \right ) &= -\frac {{\mathrm e}^{4 t} c_{1}}{3}+\frac {4 \cos \left (t \right )}{17}-\frac {2 \,{\mathrm e}^{t}}{3}-\frac {\sin \left (t \right )}{17} \\ \end{align*}

✓ Mathematica. Time used: 0.117 (sec). Leaf size: 136

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

\begin{align*} x(t)\to \int _1^t\frac {1}{4} \left (3 e^{-4 K[1]} \cos (K[1])+\cos (K[1])-6 e^{-3 K[1]}+2 e^{K[1]}\right )dK[1]-\frac {3}{4} \left (e^{4 t}-1\right ) \int _1^te^{-4 K[2]} \left (2 e^{K[2]}-\cos (K[2])\right )dK[2]-\frac {3}{4} c_2 \left (e^{4 t}-1\right )+c_1 \\ y(t)\to e^{4 t} \left (\int _1^te^{-4 K[2]} \left (2 e^{K[2]}-\cos (K[2])\right )dK[2]+c_2\right ) \\ \end{align*}

✓ Sympy. Time used: 0.470 (sec). Leaf size: 58

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

\[ \left [ x{\left (t \right )} = C_{1} - \frac {3 C_{2} e^{4 t}}{4} + e^{t} + \frac {5 \sin {\left (t \right )}}{17} - \frac {3 \cos {\left (t \right )}}{17}, \ y{\left (t \right )} = C_{2} e^{4 t} - \frac {2 e^{t}}{3} - \frac {\sin {\left (t \right )}}{17} + \frac {4 \cos {\left (t \right )}}{17}\right ] \]