Internal problem ID [3904]
Internal file name [OUTPUT/3397_Sunday_June_05_2022_09_16_07_AM_9109252/index.tex]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 23
Problem number: 657.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[_rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]
Unable to solve or complete the solution.
\[ \boxed {x \left (x^{3}-3 y x^{3}+4 y^{2}\right ) y^{\prime }-6 y^{3}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x \left (x^{3}-3 y x^{3}+4 y^{2}\right ) y^{\prime }-6 y^{3}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {6 y^{3}}{x \left (x^{3}-3 y x^{3}+4 y^{2}\right )} \end {array} \]
Maple trace
`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 Chini differential order: 1; looking for linear symmetries trying exact Looking for potential symmetries trying inverse_Riccati trying an equivalence to an Abel ODE differential order: 1; trying a linearization to 2nd order --- trying a change of variables {x -> y(x), y(x) -> x} differential order: 1; trying a linearization to 2nd order trying 1st order ODE linearizable_by_differentiation --- Trying Lie symmetry methods, 1st order --- `, `-> Computing symmetries using: way = 2 `, `-> Computing symmetries using: way = 3 `, `-> Computing symmetries using: way = 4 trying symmetry patterns for 1st order ODEs -> trying a symmetry pattern of the form [F(x)*G(y), 0] <- symmetry pattern of the form [F(x)*G(y), 0] successful`
✓ Solution by Maple
Time used: 0.046 (sec). Leaf size: 31
dsolve(x*(x^3-3*x^3*y(x)+4*y(x)^2)*diff(y(x),x) = 6*y(x)^3,y(x), singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{\operatorname {RootOf}\left (-3 x^{3} {\mathrm e}^{\textit {\_Z}}+6 c_{1} x^{3}+x^{3} \textit {\_Z} +2 \,{\mathrm e}^{2 \textit {\_Z}}\right )} \]
✓ Solution by Mathematica
Time used: 0.155 (sec). Leaf size: 27
DSolve[x(x^3-3 x^3 y[x]+4 y[x]^2)y'[x]==6 y[x]^3,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {y(x)^2}{x^3}+\frac {1}{2} (\log (y(x))-3 y(x))=c_1,y(x)\right ] \]