75.27.2 problem 777

Internal problem ID [17159]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 3 (Systems of differential equations). Section 20. The method of elimination. Exercises page 212
Problem number : 777
Date solved : Thursday, March 13, 2025 at 09:18:03 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.034 (sec). Leaf size: 36

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

\begin{align*} x \left (t \right ) &= c_{2} {\mathrm e}^{-t}+c_{1} {\mathrm e}^{t}+t -1 \\ y \left (t \right ) &= -c_{2} {\mathrm e}^{-t}+c_{1} {\mathrm e}^{t}+1-t \\ \end{align*}

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 78

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

\begin{align*} x(t)\to \frac {1}{2} e^{-t} \left (2 e^t (t-1)+(c_1+c_2) e^{2 t}+c_1-c_2\right ) \\ y(t)\to \frac {1}{2} e^{-t} \left (-2 e^t (t-1)+(c_1+c_2) e^{2 t}-c_1+c_2\right ) \\ \end{align*}

✓ Sympy. Time used: 0.116 (sec). Leaf size: 31

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

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