Internal problem ID [9042]
Internal file name [OUTPUT/7977_Monday_June_06_2022_01_04_36_AM_18034442/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 708.
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.
\[ \boxed {y^{\prime }-\frac {\left (-y^{2}+4 a x \right )^{3}}{\left (-y^{2}+4 a x -1\right ) y}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{6}-12 y^{4} a x +48 y^{2} a^{2} x^{2}-64 a^{3} x^{3}-y^{\prime } y^{3}+4 y^{\prime } y a x -y^{\prime } y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-y^{6}+12 y^{4} a x -48 y^{2} a^{2} x^{2}+64 a^{3} x^{3}}{-y^{3}+4 y a x -y} \end {array} \]
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, (-64*a^3*x^3+48*a^2*x^2*y^2-12*a*x*y^4+y^6+8*a^2*x-2*a*y^2-2*a)/(4*a*x-y^2-\
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 747
dsolve(diff(y(x),x) = (-y(x)^2+4*a*x)^3/(-y(x)^2+4*a*x-1)/y(x),y(x), singsol=all)
\[ \text {Expression too large to display} \]
✓ Solution by Mathematica
Time used: 0.273 (sec). Leaf size: 89
DSolve[y'[x] == (4*a*x - y[x]^2)^3/(y[x]*(-1 + 4*a*x - y[x]^2)),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [2 a \left (x-\frac {\text {RootSum}\left [-\text {$\#$1}^3+2 \text {$\#$1} a-2 a\&,\frac {\text {$\#$1} a \log \left (-\text {$\#$1}+4 a x-y(x)^2\right )-a \log \left (-\text {$\#$1}+4 a x-y(x)^2\right )}{2 a-3 \text {$\#$1}^2}\&\right ]}{2 a}\right )=c_1,y(x)\right ] \]