Internal problem ID [9224]
Internal file name [OUTPUT/8160_Monday_June_06_2022_01_59_13_AM_16091714/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 891.
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, [_Abel, `2nd type`, `class C`]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }-\frac {y^{2} \left (-2 y+2 x^{2}+2 x^{2} y+y x^{4}\right )}{x^{3} \left (x^{2}-y+x^{2} y\right )}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime } y x^{5}-y^{3} x^{4}+y^{\prime } x^{5}-y^{\prime } y x^{3}-2 y^{3} x^{2}-2 y^{2} x^{2}+2 y^{3}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {y^{3} x^{4}+2 y^{3} x^{2}+2 y^{2} x^{2}-2 y^{3}}{x^{5} y+x^{5}-x^{3} y} \end {array} \]
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 trying Abel <- Abel successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 56
dsolve(diff(y(x),x) = y(x)^2/x^3*(-2*y(x)+2*x^2+2*x^2*y(x)+y(x)*x^4)/(x^2-y(x)+x^2*y(x)),y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {x^{2}}{\sqrt {c_{1} -2 \ln \left (x \right )}\, x^{2}-x^{2}+1} \\ y \left (x \right ) &= -\frac {x^{2}}{\sqrt {c_{1} -2 \ln \left (x \right )}\, x^{2}+x^{2}-1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 2.31 (sec). Leaf size: 91
DSolve[y'[x] == (y[x]^2*(2*x^2 - 2*y[x] + 2*x^2*y[x] + x^4*y[x]))/(x^3*(x^2 - y[x] + x^2*y[x])),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {x^2}{1+x^2 \left (-1+\sqrt {\frac {1}{x^5}} \sqrt {x^5 (-2 \log (x)+1+c_1)}\right )} \\ y(x)\to -\frac {x^2}{-1+x^2 \left (1+\sqrt {\frac {1}{x^5}} \sqrt {x^5 (-2 \log (x)+1+c_1)}\right )} \\ y(x)\to 0 \\ \end{align*}