Internal problem ID [9231]
Internal file name [OUTPUT/8167_Monday_June_06_2022_02_00_34_AM_89934614/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 898.
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, [_1st_order, `_with_symmetry_[F(x),G(x)]`], [_Abel, `2nd type`, `class C`]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }-\frac {32 y x^{5}+8 x^{3}+32 x^{5}+64 y^{3} x^{6}+48 y^{2} x^{4}+12 y x^{2}+1}{16 x^{6} \left (4 y x^{2}+1+4 x^{2}\right )}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & -64 y^{\prime } y x^{8}+64 y^{3} x^{6}-64 y^{\prime } x^{8}-16 y^{\prime } x^{6}+48 y^{2} x^{4}+32 y x^{5}+32 x^{5}+12 y x^{2}+8 x^{3}+1=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-32 y x^{5}-8 x^{3}-32 x^{5}-64 y^{3} x^{6}-48 y^{2} x^{4}-12 y x^{2}-1}{-64 y x^{8}-64 x^{8}-16 x^{6}} \end {array} \]
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: 87
dsolve(diff(y(x),x) = 1/16/x^6*(32*x^5*y(x)+8*x^3+32*x^5+64*x^6*y(x)^3+48*x^4*y(x)^2+12*x^2*y(x)+1)/(4*x^2*y(x)+1+4*x^2),y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {4 x^{2}-\sqrt {\frac {c_{1} x +2}{x}}+1}{4 \left (\sqrt {\frac {c_{1} x +2}{x}}-1\right ) x^{2}} \\ y \left (x \right ) &= \frac {-4 x^{2}-\sqrt {\frac {c_{1} x +2}{x}}-1}{4 \left (\sqrt {\frac {c_{1} x +2}{x}}+1\right ) x^{2}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.767 (sec). Leaf size: 106
DSolve[y'[x] == (1/16 + x^3/2 + 2*x^5 + (3*x^2*y[x])/4 + 2*x^5*y[x] + 3*x^4*y[x]^2 + 4*x^6*y[x]^3)/(x^6*(1 + 4*x^2 + 4*x^2*y[x])),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {256 x^2-\sqrt {\frac {8192}{x}+c_1}+64}{4 x^2 \left (-64+\sqrt {\frac {8192}{x}+c_1}\right )} \\ y(x)\to -\frac {256 x^2+\sqrt {\frac {8192}{x}+c_1}+64}{4 x^2 \left (64+\sqrt {\frac {8192}{x}+c_1}\right )} \\ y(x)\to -\frac {1}{4 x^2} \\ \end{align*}