23.2.301 problem 321

Internal problem ID [5656]
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 : 321
Date solved : Tuesday, September 30, 2025 at 01:24:13 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: 0.038 (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: 0.005 (sec). Leaf size: 64

ode=(a^2-x^2) (D[y[x],x])^3 +b x (a^2-x^2) (D[y[x],x])^2 -D[y[x],x] -b x==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: 1.416 (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)
 

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