23.2.17 problem 17

Internal problem ID [5372]
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 : 17
Date solved : Tuesday, September 30, 2025 at 12:38:09 PM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.024 (sec). Leaf size: 45

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

\begin{align*} y &= {\mathrm e}^{\operatorname {RootOf}\left (a^{2} {\mathrm e}^{2 \textit {\_Z}} \left (\textit {\_Z}^{2}-1\right )\right )} \\ y &= {\mathrm e}^{-\sin \left (a \left (c_1 -x \right )\right )} \\ y &= {\mathrm e}^{\sin \left (a \left (c_1 -x \right )\right )} \\ \end{align*}

✓ Mathematica. Time used: 0.26 (sec). Leaf size: 83

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

\begin{align*} y(x)&\to e^{\frac {1}{2} \left (e^{i a x-c_1}+e^{c_1-i a x}\right )}\\ y(x)&\to e^{\frac {1}{2} \left (e^{-i a x-c_1}+e^{i a x+c_1}\right )}\\ y(x)&\to \frac {1}{e}\\ y(x)&\to e \end{align*}

✓ Sympy. Time used: 2.782 (sec). Leaf size: 51

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

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