23.2.206 problem 211

Internal problem ID [5561]
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 : 211
Date solved : Tuesday, September 30, 2025 at 12:53:17 PM
CAS classification : [_quadrature]

\begin{align*} \left (1-a y\right ) {y^{\prime }}^{2}&=a y \end{align*}

✓ Maple. Time used: 0.644 (sec). Leaf size: 399

ode:=(1-a*y(x))*diff(y(x),x)^2 = a*y(x); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} \text {Solution too large to show}\end{align*}

✓ Mathematica. Time used: 0.553 (sec). Leaf size: 115

ode=(1-a y[x]) (D[y[x],x])^2==a y[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \text {InverseFunction}\left [\frac {\arcsin \left (\sqrt {\text {$\#$1}} \sqrt {a}\right )}{\sqrt {a}}+\sqrt {\text {$\#$1}} \sqrt {1-\text {$\#$1} a}\&\right ]\left [-\sqrt {a} x+c_1\right ]\\ y(x)&\to \text {InverseFunction}\left [\frac {\arcsin \left (\sqrt {\text {$\#$1}} \sqrt {a}\right )}{\sqrt {a}}+\sqrt {\text {$\#$1}} \sqrt {1-\text {$\#$1} a}\&\right ]\left [\sqrt {a} x+c_1\right ]\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 0.793 (sec). Leaf size: 41

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(-a*y(x) + (-a*y(x) + 1)*Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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