Internal problem ID [11333]
Internal file name [OUTPUT/10319_Tuesday_December_27_2022_04_07_34_AM_59523079/index.tex]
Book: An elementary treatise on differential equations by Abraham Cohen. DC heath publishers.
1906
Section: Chapter IX, Miscellaneous methods for solving equations of higher order than first.
Article 61. Transformation of variables. Page 143
Problem number: Ex 3.
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_2nd_order, _reducible, _mu_xy]]
Unable to solve or complete the solution.
\[ \boxed {y y^{\prime \prime }-{y^{\prime }}^{2}-y^{2} \ln \left (y\right )+x^{2} y^{2}=0} \]
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying 2nd order Liouville trying 2nd order WeierstrassP trying 2nd order JacobiSN differential order: 2; trying a linearization to 3rd order trying 2nd order ODE linearizable_by_differentiation trying 2nd order, 2 integrating factors of the form mu(x,y) trying a quadrature checking if the LODE has constant coefficients <- constant coefficients successful <- 2nd order, 2 integrating factors of the form mu(x,y) successful`
✓ Solution by Maple
Time used: 0.046 (sec). Leaf size: 22
dsolve(y(x)*diff(y(x),x$2)-diff(y(x),x)^2=y(x)^2*ln(y(x))-x^2*y(x)^2,y(x), singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{x^{2}+2-\frac {c_{2} {\mathrm e}^{x}}{2}+\frac {c_{1} {\mathrm e}^{-x}}{2}} \]
✓ Solution by Mathematica
Time used: 1.156 (sec). Leaf size: 30
DSolve[y[x]*y''[x]-y'[x]^2==y[x]^2*Log[y[x]]-x^2*y[x]^2,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{x^2-\frac {c_1 e^x}{2}-c_2 e^{-x}+2} \]