69.1.119 problem 170

Internal problem ID [14193]
Book : DIFFERENTIAL and INTEGRAL CALCULUS. VOL I. by N. PISKUNOV. MIR PUBLISHERS, Moscow 1969.
Section : Chapter 8. Differential equations. Exercises page 595
Problem number : 170
Date solved : Wednesday, March 05, 2025 at 10:39:30 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=y \left (t \right )+1\\ \frac {d}{d t}y \left (t \right )&=x \left (t \right )+1 \end{align*}

With initial conditions

\begin{align*} x \left (0\right ) = -2\\ y \left (0\right ) = 0 \end{align*}

✓ Maple. Time used: 0.038 (sec). Leaf size: 21

ode:=[diff(x(t),t) = y(t)+1, diff(y(t),t) = x(t)+1]; 
ic:=x(0) = -2y(0) = 0; 
dsolve([ode,ic]);
 

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

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 24

ode={D[x[t],t]==y[t]+1,D[y[t],t]==x[t]+1}; 
ic={x[0]==-2,y[0]==0}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to -e^{-t}-1 \\ y(t)\to e^{-t}-1 \\ \end{align*}

✓ Sympy. Time used: 0.134 (sec). Leaf size: 27

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-y(t) + Derivative(x(t), t) - 1,0),Eq(-x(t) + Derivative(y(t), t) - 1,0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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