1.87 problem 86

Internal problem ID [7131]
Internal file name [OUTPUT/6117_Sunday_June_05_2022_04_23_26_PM_23415603/index.tex]

Book: Own collection of miscellaneous problems
Section: section 1.0
Problem number: 86.
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, _exact, _nonlinear], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]]

Unable to solve or complete the solution.

\[ \boxed {y^{\prime \prime }-\frac {1}{y}+\frac {x y^{\prime }}{y^{2}}=0} \]

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 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) 
trying differential order: 2; exact nonlinear 
-> Calling odsolve with the ODE`, diff(_b(_a), _a) = -(c__1*_b(_a)-_a)/_b(_a), _b(_a)`   *** Sublevel 2 *** 
   Methods for first order ODEs: 
   --- Trying classification methods --- 
   trying a quadrature 
   trying 1st order linear 
   trying Bernoulli 
   trying separable 
   trying inverse linear 
   trying homogeneous types: 
   trying homogeneous D 
   <- homogeneous successful 
<- differential order: 2; exact nonlinear successful`
 

Solution by Maple

Time used: 0.031 (sec). Leaf size: 56

dsolve(diff(y(x),x$2)=1/y(x)-x/y(x)^2*diff(y(x),x),y(x), singsol=all)
 

\[ y \left (x \right ) = \operatorname {RootOf}\left (\textit {\_Z}^{2}-{\mathrm e}^{\operatorname {RootOf}\left (x^{2} \left (4 \,{\mathrm e}^{\textit {\_Z}} {\cosh \left (\frac {\sqrt {c_{1}^{2}+4}\, \left (2 c_{2} +\textit {\_Z} +2 \ln \left (x \right )\right )}{2 c_{1}}\right )}^{2}+c_{1}^{2}+4\right )\right )}-1+\textit {\_Z} c_{1} \right ) x \]

Solution by Mathematica

Time used: 0.199 (sec). Leaf size: 77

DSolve[y''[x]==1/y[x]-x/y[x]^2*y'[x],y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\frac {1}{2} \log \left (-\frac {y(x)^2}{x^2}-\frac {c_1 y(x)}{x}+1\right )-\frac {c_1 \arctan \left (\frac {\frac {2 y(x)}{x}+c_1}{\sqrt {-4-c_1{}^2}}\right )}{\sqrt {-4-c_1{}^2}}=-\log (x)+c_2,y(x)\right ] \]