Internal problem ID [3929]
Internal file name [OUTPUT/3422_Sunday_June_05_2022_09_18_14_AM_39052743/index.tex]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 24
Problem number: 682.
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]
Unable to solve or complete the solution.
\[ \boxed {\left (a -3 x^{2}-y^{2}\right ) y y^{\prime }+x \left (a -x^{2}+y^{2}\right )=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (a -3 x^{2}-y^{2}\right ) y y^{\prime }+x \left (a -x^{2}+y^{2}\right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {x \left (a -x^{2}+y^{2}\right )}{\left (a -3 x^{2}-y^{2}\right ) y} \end {array} \]
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`[0, (x^4+2*x^2*y^2+y^4)/(-3*x^2-y^2+a)/y]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 122
dsolve((a-3*x^2-y(x)^2)*y(x)*diff(y(x),x)+x*(a-x^2+y(x)^2) = 0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {\sqrt {-\operatorname {LambertW}\left (-\left (-2 x^{2}+a \right ) {\mathrm e}^{2} c_{1} \right ) \left (x^{2} \operatorname {LambertW}\left (-\left (-2 x^{2}+a \right ) {\mathrm e}^{2} c_{1} \right )-2 x^{2}+a \right )}}{\operatorname {LambertW}\left (-\left (-2 x^{2}+a \right ) {\mathrm e}^{2} c_{1} \right )} \\ y \left (x \right ) &= -\frac {\sqrt {-\operatorname {LambertW}\left (-\left (-2 x^{2}+a \right ) {\mathrm e}^{2} c_{1} \right ) \left (x^{2} \operatorname {LambertW}\left (-\left (-2 x^{2}+a \right ) {\mathrm e}^{2} c_{1} \right )-2 x^{2}+a \right )}}{\operatorname {LambertW}\left (-\left (-2 x^{2}+a \right ) {\mathrm e}^{2} c_{1} \right )} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.336 (sec). Leaf size: 39
DSolve[(a-3*x^2-y[x]^2)*y[x]*y'[x]+x*(a-x^2+y[x]^2)==0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {1}{2} \left (\frac {a+2 y(x)^2}{x^2+y(x)^2}+\log \left (x^2+y(x)^2\right )\right )=c_1,y(x)\right ] \]