82.8.30 problem 36-29 (b)

Internal problem ID [21901]
Book : The Differential Equations Problem Solver. VOL. II. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 36. Nonlinear differential equations. Page 1203
Problem number : 36-29 (b)
Date solved : Sunday, October 12, 2025 at 05:51:37 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.048 (sec). Leaf size: 39

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

\begin{align*} \left \{x \left (t \right ) &= \frac {1}{-t +c_2}\right \} \\ \left \{y \left (t \right ) &= \frac {{\mathrm e}^{\int -x \left (t \right )d t}}{\int -2 \,{\mathrm e}^{\int -x \left (t \right )d t}d t +c_1}\right \} \\ \end{align*}

✓ Mathematica. Time used: 0.041 (sec). Leaf size: 37

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

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

✓ Sympy. Time used: 0.154 (sec). Leaf size: 26

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

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