Internal problem ID [9074]
Internal file name [OUTPUT/8009_Monday_June_06_2022_01_15_14_AM_92228732/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 740.
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.
\[ \boxed {y^{\prime }-\frac {x +y^{4}-2 y^{2} x^{2}+x^{4}}{y}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & -y^{4}+2 y^{2} x^{2}-x^{4}+y y^{\prime }-x =0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {x +y^{4}-2 y^{2} x^{2}+x^{4}}{y} \end {array} \]
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) = -(4*x^4+4*x)*y(x)-4*x^2*(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 Reducible group (found an exponential solution) <- Kovacics algorithm successful <- differential order: 1; linearization to 2nd order successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 72
dsolve(diff(y(x),x) = (x+y(x)^4-2*x^2*y(x)^2+x^4)/y(x),y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -\frac {\sqrt {2}\, \sqrt {\left (x +c_{1} \right ) \left (2 c_{1} x^{2}+2 x^{3}-1\right )}}{2 c_{1} +2 x} \\ y \left (x \right ) &= \frac {\sqrt {2}\, \sqrt {\left (x +c_{1} \right ) \left (2 c_{1} x^{2}+2 x^{3}-1\right )}}{2 c_{1} +2 x} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.661 (sec). Leaf size: 132
DSolve[y'[x] == (x + x^4 - 2*x^2*y[x]^2 + y[x]^4)/y[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sqrt {2 x^3+2 c_1 x^2-1}}{\sqrt {2} \sqrt {x+c_1}} \\ y(x)\to \frac {\sqrt {2 x^3+2 c_1 x^2-1}}{\sqrt {2} \sqrt {x+c_1}} \\ y(x)\to -i \sqrt {-x^2} \\ y(x)\to i \sqrt {-x^2} \\ y(x)\to -\sqrt {x^2} \\ y(x)\to \sqrt {x^2} \\ \end{align*}