Internal problem ID [12255]
Internal file name [OUTPUT/10908_Thursday_September_28_2023_01_08_28_AM_72433297/index.tex]
Book: APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A.
Dobrushkin. CRC Press 2015
Section: Chapter 4, Second and Higher Order Linear Differential Equations. Problems page
221
Problem number: Problem 15.
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime \prime }+\frac {k x}{y^{4}}=0} \]
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 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 trying 2nd order, integrating factor of the form mu(y) trying 2nd order, integrating factor of the form mu(x,y) trying 2nd order, integrating factor of the form mu(x,y)/(y)^n, only the general case trying 2nd order, integrating factor of the form mu(y,y) -> Calling odsolve with the ODE`, -(_y1^3*k+16*x^2)*y(x)/(k*_y1^3)-(4/3)*x^2*(3*(diff(y(x), x))*x+5*_y1)/(k*_y1^3), y(x)` *** Subl Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful trying differential order: 2; mu polynomial in y trying 2nd order, integrating factor of the form mu(x,y) differential order: 2; looking for linear symmetries differential order: 2; found: 1 linear symmetries. Trying reduction of order `, `2nd order, trying reduction of order with given symmetries:`[5/3*x, y]
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 97
dsolve(diff(y(x),x$2)+k*x/(y(x)^4)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \operatorname {RootOf}\left (15 \sqrt {3}\, \left (\int _{}^{\textit {\_Z}}\frac {\sqrt {-\textit {\_f}^{4} c_{1} +50 \textit {\_f} k}\, \textit {\_f}}{\textit {\_f}^{3} c_{1} -50 k}d \textit {\_f} \right ) x -5 c_{2} x -3\right ) x \\ y \left (x \right ) &= \operatorname {RootOf}\left (15 \sqrt {3}\, \left (\int _{}^{\textit {\_Z}}\frac {\sqrt {-\textit {\_f}^{4} c_{1} +50 \textit {\_f} k}\, \textit {\_f}}{\textit {\_f}^{3} c_{1} -50 k}d \textit {\_f} \right ) x +5 c_{2} x +3\right ) x \\ \end{align*}
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y''[x]+k*x/(y[x]^4)==0,y[x],x,IncludeSingularSolutions -> True]
Not solved