82.44.3 problem Ex. 3

Internal problem ID [18967]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter VIII. Exact differential equations, and equations of particular forms. Integration in series. problems at page 99
Problem number : Ex. 3
Date solved : Tuesday, January 28, 2025 at 12:40:59 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]]

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

✓ Solution by Maple

Time used: 0.026 (sec). Leaf size: 18

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

\[ y \left (x \right ) = {\mathrm e}^{\frac {{\mathrm e}^{-x} c_{1}}{2}-\frac {c_{2} {\mathrm e}^{x}}{2}} \]

✓ Solution by Mathematica

Time used: 0.958 (sec). Leaf size: 63

DSolve[y[x]*D[y[x],{x,2}]-D[y[x],x]^2==y[x]^2*Log[y[x]],y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to e^{\frac {1}{2} \left (e^{x+c_2}-c_1 e^{-x-c_2}\right )} \\ y(x)\to e^{\frac {1}{2} \left (e^{-x-c_2}-c_1 e^{x+c_2}\right )} \\ \end{align*}