54.1.523 problem 536

Internal problem ID [11837]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 536
Date solved : Tuesday, September 30, 2025 at 11:16:17 PM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.031 (sec). Leaf size: 52

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

\begin{align*} y &= -\frac {b \,x^{2}}{2}+c_1 \\ y &= \arctan \left (\frac {x}{\sqrt {a^{2}-x^{2}}}\right )+c_1 \\ y &= -\arctan \left (\frac {x}{\sqrt {a^{2}-x^{2}}}\right )+c_1 \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 64

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

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

✓ Sympy. Time used: 1.389 (sec). Leaf size: 48

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

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