29.32.24 problem 958

Internal problem ID [5536]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 32
Problem number : 958
Date solved : Sunday, March 30, 2025 at 08:30:00 AM
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: 0.911 (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: 1.258 (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)
 

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