83.23.1 problem 1

Internal problem ID [19211]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter V. Singular solutions. Exercise V at page 76
Problem number : 1
Date solved : Monday, March 31, 2025 at 06:57:23 PM
CAS classification : [_rational, _Bernoulli]

\begin{align*} x^{2} y^{2}-3 x y y^{\prime }&=2 y^{2}+x^{3} \end{align*}

✓ Maple. Time used: 0.010 (sec). Leaf size: 144

ode:=x^2*y(x)^2-3*x*y(x)*diff(y(x),x) = 2*y(x)^2+x^3; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -\frac {\sqrt {\frac {{\mathrm e}^{-\frac {2 x^{2}}{3}} x^{{4}/{3}} \left (-3 \,3^{{1}/{12}} x^{{1}/{3}} \operatorname {WhittakerM}\left (\frac {1}{12}, \frac {7}{12}, \frac {x^{2}}{3}\right ) {\mathrm e}^{\frac {x^{2}}{6}}+\left (x^{{7}/{3}}+{\mathrm e}^{\frac {x^{2}}{3}} c_1 \right ) \left (x^{2}\right )^{{1}/{12}}\right )}{\left (x^{2}\right )^{{1}/{12}}}}\, {\mathrm e}^{\frac {x^{2}}{3}}}{x^{{4}/{3}}} \\ y &= \frac {\sqrt {\frac {{\mathrm e}^{-\frac {2 x^{2}}{3}} x^{{4}/{3}} \left (-3 \,3^{{1}/{12}} x^{{1}/{3}} \operatorname {WhittakerM}\left (\frac {1}{12}, \frac {7}{12}, \frac {x^{2}}{3}\right ) {\mathrm e}^{\frac {x^{2}}{6}}+\left (x^{{7}/{3}}+{\mathrm e}^{\frac {x^{2}}{3}} c_1 \right ) \left (x^{2}\right )^{{1}/{12}}\right )}{\left (x^{2}\right )^{{1}/{12}}}}\, {\mathrm e}^{\frac {x^{2}}{3}}}{x^{{4}/{3}}} \\ \end{align*}

✓ Mathematica. Time used: 7.582 (sec). Leaf size: 113

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

\begin{align*} y(x)\to -\sqrt {\frac {e^{\frac {x^2}{3}} \left (3 \sqrt [6]{3} \left (x^2\right )^{5/6} \Gamma \left (\frac {13}{6},\frac {x^2}{3}\right )+c_1 x^{5/3}\right )}{x^3}} \\ y(x)\to \sqrt {\frac {e^{\frac {x^2}{3}} \left (3 \sqrt [6]{3} \left (x^2\right )^{5/6} \Gamma \left (\frac {13}{6},\frac {x^2}{3}\right )+c_1 x^{5/3}\right )}{x^3}} \\ \end{align*}

✗ Sympy

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

Timed Out