60.1.372 problem 373

Internal problem ID [10386]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 373
Date solved : Monday, January 27, 2025 at 07:38:47 PM
CAS classification : [_quadrature]

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

✓ Solution by Maple

Time used: 0.069 (sec). Leaf size: 45

dsolve(diff(y(x),x)^2+a^2*y(x)^2*(ln(y(x))^2-1) = 0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.452 (sec). Leaf size: 83

DSolve[a^2*(-1 + Log[y[x]]^2)*y[x]^2 + D[y[x],x]^2==0,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*}