Internal problem ID [8976]
Internal file name [OUTPUT/7911_Monday_June_06_2022_12_54_17_AM_36149538/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 642.
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.
\[ \boxed {y^{\prime }-\frac {\left (-y^{2}+4 x a \right )^{2}}{y}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & -y^{4}+8 a y^{2} x -16 a^{2} x^{2}+y y^{\prime }=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {y^{4}-8 a y^{2} x +16 a^{2} x^{2}}{y} \end {array} \]
Maple trace Kovacic algorithm successful
`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 -> Calling odsolve with the ODE`, diff(diff(y(x), x), x) = -64*a^2*x^2*y(x)-16*a*x*(diff(y(x), x)), y(x)` *** Sublevel 2 *** Methods for second order ODEs: --- Trying classification methods --- trying a quadrature checking if the LODE has constant coefficients checking if the LODE is of Euler type trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Trying a Liouvillian solution using Kovacics algorithm A Liouvillian solution exists Group is reducible or imprimitive <- Kovacics algorithm successful <- differential order: 1; linearization to 2nd order successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 227
dsolve(diff(y(x),x) = (-y(x)^2+4*a*x)^2/y(x),y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {2 \,{\mathrm e}^{4 a \,x^{2}+2 \sqrt {2}\, \sqrt {a}\, x} \sqrt {\sqrt {a}\, \left (c_{1} {\mathrm e}^{4 \sqrt {2}\, \sqrt {a}\, x}+1\right ) \left (c_{1} \left (x \sqrt {a}-\frac {\sqrt {2}}{4}\right ) {\mathrm e}^{4 \sqrt {2}\, \sqrt {a}\, x}+x \sqrt {a}+\frac {\sqrt {2}}{4}\right ) {\mathrm e}^{-8 a \,x^{2}} {\mathrm e}^{-4 \sqrt {2}\, \sqrt {a}\, x}}}{c_{1} {\mathrm e}^{4 \sqrt {2}\, \sqrt {a}\, x}+1} \\ y \left (x \right ) &= -\frac {2 \,{\mathrm e}^{4 a \,x^{2}+2 \sqrt {2}\, \sqrt {a}\, x} \sqrt {\sqrt {a}\, \left (c_{1} {\mathrm e}^{4 \sqrt {2}\, \sqrt {a}\, x}+1\right ) \left (c_{1} \left (x \sqrt {a}-\frac {\sqrt {2}}{4}\right ) {\mathrm e}^{4 \sqrt {2}\, \sqrt {a}\, x}+x \sqrt {a}+\frac {\sqrt {2}}{4}\right ) {\mathrm e}^{-8 a \,x^{2}} {\mathrm e}^{-4 \sqrt {2}\, \sqrt {a}\, x}}}{c_{1} {\mathrm e}^{4 \sqrt {2}\, \sqrt {a}\, x}+1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 23.997 (sec). Leaf size: 95
DSolve[y'[x] == (4*a*x - y[x]^2)^2/y[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\sqrt {4 a x-\sqrt {2} \sqrt {a} \tanh \left (\frac {\sqrt {2} (2 a x-c_1)}{\sqrt {a}}\right )} \\ y(x)\to \sqrt {4 a x-\sqrt {2} \sqrt {a} \tanh \left (\frac {\sqrt {2} (2 a x-c_1)}{\sqrt {a}}\right )} \\ \end{align*}