82.8.29 problem 36-29 (a)

Internal problem ID [21900]
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 (a)
Date solved : Sunday, October 12, 2025 at 05:51:36 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=2 x \left (t \right ) y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=3 y \left (t \right )^{2}-x \left (t \right )^{2} \end{align*}

✓ Maple. Time used: 0.384 (sec). Leaf size: 62

ode:=[diff(x(t),t) = 2*x(t)*y(t), diff(y(t),t) = 3*y(t)^2-x(t)^2]; 
dsolve(ode);
 

\begin{align*} \left [\left \{\frac {d^{2}}{d t^{2}}x \left (t \right ) &= \frac {-4 x \left (t \right )^{4}+5 \left (\frac {d}{d t}x \left (t \right )\right )^{2}}{2 x \left (t \right )}\right \}, \left \{y \left (t \right ) = \frac {\frac {d}{d t}x \left (t \right )}{2 x \left (t \right )}\right \}\right ] \\ \left [\{x \left (t \right ) = 0\}, \left \{y \left (t \right ) &= \frac {1}{-3 t +c_1}\right \}\right ] \\ \end{align*}

✓ Mathematica. Time used: 0.071 (sec). Leaf size: 282

ode={D[x[t],t]==2*x[t]*y[t],D[y[t],t]==3*y[t]^2-x[t]^2}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to -\text {InverseFunction}\left [c_1 \text {arctanh}\left (\sqrt {1+\text {$\#$1} c_1}\right )-\frac {\sqrt {1+\text {$\#$1} c_1}}{\text {$\#$1}}\&\right ][-2 t+c_2] \sqrt {1+c_1 \text {InverseFunction}\left [c_1 \text {arctanh}\left (\sqrt {1+\text {$\#$1} c_1}\right )-\frac {\sqrt {1+\text {$\#$1} c_1}}{\text {$\#$1}}\&\right ][-2 t+c_2]}\\ x(t)&\to \text {InverseFunction}\left [c_1 \text {arctanh}\left (\sqrt {1+\text {$\#$1} c_1}\right )-\frac {\sqrt {1+\text {$\#$1} c_1}}{\text {$\#$1}}\&\right ][-2 t+c_2]\\ y(t)&\to \text {InverseFunction}\left [c_1 \text {arctanh}\left (\sqrt {1+\text {$\#$1} c_1}\right )-\frac {\sqrt {1+\text {$\#$1} c_1}}{\text {$\#$1}}\&\right ][2 t+c_2] \sqrt {1+c_1 \text {InverseFunction}\left [c_1 \text {arctanh}\left (\sqrt {1+\text {$\#$1} c_1}\right )-\frac {\sqrt {1+\text {$\#$1} c_1}}{\text {$\#$1}}\&\right ][2 t+c_2]}\\ x(t)&\to \text {InverseFunction}\left [c_1 \text {arctanh}\left (\sqrt {1+\text {$\#$1} c_1}\right )-\frac {\sqrt {1+\text {$\#$1} c_1}}{\text {$\#$1}}\&\right ][2 t+c_2] \end{align*}

✗ Sympy

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-2*x(t)*y(t) + Derivative(x(t), t),0),Eq(x(t)**2 - 3*y(t)**2 + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

NotImplementedError : multiple generators [asinh(1/(sqrt(C1)*sqrt(u))), sqrt(1 + 1/(C1*u)), sqrt(u)] 
No algorithms are implemented to solve equation sqrt(C1)*sqrt(1 + 1/(C1*u))/(2*sqrt(u)) - C1*asinh(1/(sqrt(C1)*sqrt(u)))/2 - C2 - t