29.26.28 problem 764

Internal problem ID [5349]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 26
Problem number : 764
Date solved : Monday, January 27, 2025 at 11:16:48 AM
CAS classification : [_quadrature]

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

✓ Solution by Maple

Time used: 0.075 (sec). Leaf size: 45

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

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

✓ Solution by Mathematica

Time used: 0.458 (sec). Leaf size: 83

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