64.3.11 problem 12

Internal problem ID [13211]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 2, section 2.1 (Exact differential equations and integrating factors). Exercises page 37
Problem number : 12
Date solved : Monday, March 31, 2025 at 07:38:00 AM
CAS classification : [_exact, _rational]

\begin{align*} 3 x^{2} y^{2}-y^{3}+2 x +\left (2 x^{3} y-3 x y^{2}+1\right ) y^{\prime }&=0 \end{align*}

With initial conditions

\begin{align*} y \left (-2\right )&=1 \end{align*}

✓ Maple. Time used: 1.436 (sec). Leaf size: 208

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

\[ y = \frac {-\frac {\left (1+i \sqrt {3}\right ) 2^{{2}/{3}} {\left (\left (2 x^{7}+3 \sqrt {3}\, \sqrt {\frac {4 x^{10}+4 x^{8}+44 x^{5}+72 x^{3}+27 x -4}{x}}+36 x^{2}+27\right ) x^{2}\right )}^{{2}/{3}}}{2}+\left (2 x^{2} {\left (\left (2 x^{7}+3 \sqrt {3}\, \sqrt {\frac {4 x^{10}+4 x^{8}+44 x^{5}+72 x^{3}+27 x -4}{x}}+36 x^{2}+27\right ) x^{2}\right )}^{{1}/{3}}+\left (i \sqrt {3}-1\right ) 2^{{1}/{3}} \left (x^{5}+3\right )\right ) x}{6 {\left (\left (2 x^{7}+3 \sqrt {3}\, \sqrt {\frac {4 x^{10}+4 x^{8}+44 x^{5}+72 x^{3}+27 x -4}{x}}+36 x^{2}+27\right ) x^{2}\right )}^{{1}/{3}} x} \]

✓ Mathematica. Time used: 60.242 (sec). Leaf size: 250

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

\[ y(x)\to \frac {2 \sqrt [3]{2} \left (1-i \sqrt {3}\right ) x^6+4 \sqrt [3]{-2 x^9-36 x^4-27 x^2+3 \sqrt {3} \sqrt {x^3 \left (4 x^{10}+4 x^8+44 x^5+72 x^3+27 x-4\right )}} x^3+\left (1+i \sqrt {3}\right ) \left (-4 x^9-72 x^4-54 x^2+6 \sqrt {3} \sqrt {x^3 \left (4 x^{10}+4 x^8+44 x^5+72 x^3+27 x-4\right )}\right )^{2/3}+6 \sqrt [3]{2} \left (1-i \sqrt {3}\right ) x}{12 x \sqrt [3]{-2 x^9-36 x^4-27 x^2+3 \sqrt {3} \sqrt {x^3 \left (4 x^{10}+4 x^8+44 x^5+72 x^3+27 x-4\right )}}} \]

✗ Sympy

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

Timed Out