89.2.33 problem 35

Internal problem ID [24298]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 2. Equations of the first order and first degree. Exercises at page 27
Problem number : 35
Date solved : Thursday, October 02, 2025 at 10:10:52 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

\begin{align*} -2 y x +\left (3 x^{2}-2 y^{2}\right ) y^{\prime }&=0 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.176 (sec). Leaf size: 106

ode:=-2*x*y(x)+(3*x^2-2*y(x)^2)*diff(y(x),x) = 0; 
ic:=[y(0) = -1]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {\left (i \sqrt {3}-1\right ) \left (54 x^{2}-8+6 \sqrt {3}\, \sqrt {27 x^{4}-8 x^{2}}\right )^{{2}/{3}}-4 i \sqrt {3}-4 \left (54 x^{2}-8+6 \sqrt {3}\, \sqrt {27 x^{4}-8 x^{2}}\right )^{{1}/{3}}-4}{12 \left (54 x^{2}-8+6 \sqrt {3}\, \sqrt {27 x^{4}-8 x^{2}}\right )^{{1}/{3}}} \]

✓ Mathematica. Time used: 60.207 (sec). Leaf size: 189

ode=-2*x*y[x]+( 3*x^2-2*y[x]^2 )*D[y[x],x]==0; 
ic={y[0]==-1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {i \left (\sqrt [3]{2} \sqrt {3} \left (27 x^2+3 \sqrt {3} \sqrt {x^2 \left (27 x^2-8\right )}-4\right )^{2/3}+i \sqrt [3]{2} \left (27 x^2+3 \sqrt {3} \sqrt {x^2 \left (27 x^2-8\right )}-4\right )^{2/3}+4 i \sqrt [3]{27 x^2+3 \sqrt {3} \sqrt {x^2 \left (27 x^2-8\right )}-4}-2\ 2^{2/3} \sqrt {3}+2 i 2^{2/3}\right )}{12 \sqrt [3]{27 x^2+3 \sqrt {3} \sqrt {x^2 \left (27 x^2-8\right )}-4}} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-2*x*y(x) + (3*x**2 - 2*y(x)**2)*Derivative(y(x), x),0) 
ics = {y(0): -1} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out