54.1.364 problem 373

Internal problem ID [11678]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 373
Date solved : Tuesday, September 30, 2025 at 10:06:23 PM
CAS classification : [_quadrature]

\begin{align*} {y^{\prime }}^{2}+a^{2} y^{2} \left (\ln \left (y\right )^{2}-1\right )&=0 \end{align*}

✓ Maple. Time used: 0.025 (sec). Leaf size: 45

ode:=diff(y(x),x)^2+a^2*y(x)^2*(ln(y(x))^2-1) = 0; 
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: 0.275 (sec). Leaf size: 83

ode=a^2*(-1 + Log[y[x]]^2)*y[x]^2 + D[y[x],x]^2==0; 
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: 2.774 (sec). Leaf size: 51

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

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