87.29.7 problem 7

Internal problem ID [23887]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 8. Nonlinear differential equations and systems. Exercise at page 304
Problem number : 7
Date solved : Thursday, October 02, 2025 at 09:46:17 PM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} x \left (0\right )&=1 \\ y \left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.051 (sec). Leaf size: 13

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

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

✓ Mathematica. Time used: 0.034 (sec). Leaf size: 16

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

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

✓ Sympy. Time used: 0.047 (sec). Leaf size: 12

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

\[ \left [ x{\left (t \right )} = e^{t}, \ y{\left (t \right )} = t e^{t}\right ] \]