Internal problem ID [7477]
Internal file name [OUTPUT/6444_Sunday_June_19_2022_05_02_47_PM_12898953/index.tex]
Book: Second order enumerated odes
Section: section 2
Problem number: 36.
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, _with_linear_symmetries]]
Unable to solve or complete the solution.
\[ \boxed {x^{2} y^{\prime \prime }+\left (y^{\prime } x -y\right )^{2}=0} \]
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 differential order: 2; missing variables -> trying 2nd order, dynamical_symmetries, fully reducible to Abel through one integrating factor of the form G(x,y)/(1+H(x,y)*y)^2 --- trying a change of variables {x -> y(x), y(x) -> x} and re-entering methods for dynamical symmetries --- -> trying 2nd order, dynamical_symmetries, fully reducible to Abel through one integrating factor of the form G(x,y)/(1+H(x,y)*y) trying 2nd order, integrating factors of the form mu(x,y)/(y)^n, only the singular cases trying symmetries linear in x and y(x) <- linear symmetries successful`
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 22
dsolve(x^2*diff(y(x),x$2)+(x*diff(y(x),x)-y(x))^2=0,y(x), singsol=all)
\[ y \left (x \right ) = \left (-{\mathrm e}^{c_{1}} \operatorname {expIntegral}_{1}\left (-\ln \left (\frac {1}{x}\right )+c_{1} \right )+c_{2} \right ) x \]
✓ Solution by Mathematica
Time used: 46.789 (sec). Leaf size: 33
DSolve[x^2*y''[x]+(x*y'[x]-y[x])^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to x \left (e^{c_1} \operatorname {ExpIntegralEi}(-c_1-\log (x))+c_2\right ) \\ y(x)\to c_2 x \\ \end{align*}