Internal problem ID [9148]
Internal file name [OUTPUT/8083_Monday_June_06_2022_01_42_07_AM_45123567/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 814.
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`], [_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }-\frac {y \left (-3 y x^{3}-3+y^{2} x^{7}\right )}{x \left (y x^{3}+1\right )}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{3} x^{7}-y^{\prime } y x^{4}-3 y^{2} x^{3}-y^{\prime } x -3 y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-y^{3} x^{7}+3 y^{2} x^{3}+3 y}{-y x^{4}-x} \end {array} \]
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: 38
dsolve(diff(y(x),x) = y(x)/x*(-3*x^3*y(x)-3+y(x)^2*x^7)/(x^3*y(x)+1),y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {1}{\left (\sqrt {-2 x +c_{1}}-1\right ) x^{3}} \\ y \left (x \right ) &= -\frac {1}{\left (\sqrt {-2 x +c_{1}}+1\right ) x^{3}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.776 (sec). Leaf size: 75
DSolve[y'[x] == (y[x]*(-3 - 3*x^3*y[x] + x^7*y[x]^2))/(x*(1 + x^3*y[x])),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {x}{-x^4+\frac {\sqrt {x (-2 x+1+c_1)}}{\sqrt {\frac {1}{x^7}}}} \\ y(x)\to -\frac {x}{x^4+\frac {\sqrt {x (-2 x+1+c_1)}}{\sqrt {\frac {1}{x^7}}}} \\ y(x)\to 0 \\ \end{align*}