60.1.433 problem 445

Internal problem ID [10447]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 445
Date solved : Sunday, March 30, 2025 at 04:45:01 PM
CAS classification : [_quadrature]

\begin{align*} x^{2} {y^{\prime }}^{2}+\left (a \,x^{2} y^{3}+b \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*x^2*y(x)^3+b)*diff(y(x),x)+a*b*y(x)^3 = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.07 (sec). Leaf size: 49

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

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

✓ Sympy. Time used: 0.944 (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*x**2*y(x)**3 + b)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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