23.2.163 problem 166

Internal problem ID [5518]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 2. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF SECOND OR HIGHER DEGREE, page 278
Problem number : 166
Date solved : Tuesday, September 30, 2025 at 12:50:23 PM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 35

ode:=x^2*diff(y(x),x)^2+(a+b*x^2*y(x)^3)*diff(y(x),x)+a*b*y(x)^3 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {a}{x}+c_1 \\ y &= \frac {1}{\sqrt {2 b x +c_1}} \\ y &= -\frac {1}{\sqrt {2 b x +c_1}} \\ \end{align*}

✓ Mathematica. Time used: 0.043 (sec). Leaf size: 49

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

\begin{align*} y(x)&\to -\frac {1}{\sqrt {2 b x-2 c_1}}\\ y(x)&\to \frac {1}{\sqrt {2 b x-2 c_1}}\\ y(x)&\to \frac {a}{x}+c_1 \end{align*}

✓ Sympy. Time used: 0.645 (sec). Leaf size: 49

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

\[ \left [ y{\left (x \right )} = C_{1} + \frac {a}{x}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {- \frac {1}{C_{1} - b x}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {- \frac {1}{C_{1} - b x}}}{2}\right ] \]