Internal problem ID [9020]
Internal file name [OUTPUT/7955_Monday_June_06_2022_01_00_51_AM_22468490/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 686.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_Abel, `2nd type`, `class C`], [_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }-\frac {y^{3} x \,{\mathrm e}^{2 x^{2}}}{y \,{\mathrm e}^{x^{2}}+1}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & -y^{3} x \,{\mathrm e}^{2 x^{2}}+y^{\prime } y \,{\mathrm e}^{x^{2}}+y^{\prime }=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {y^{3} x \,{\mathrm e}^{2 x^{2}}}{y \,{\mathrm e}^{x^{2}}+1} \end {array} \]
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: 0.016 (sec). Leaf size: 47
dsolve(diff(y(x),x) = y(x)^3/(y(x)*exp(x^2)+1)*x*exp(2*x^2),y(x), singsol=all)
\[ y \left (x \right ) = \left (\cot \left (\operatorname {RootOf}\left (-2 x^{2}-\ln \left (2\right )+\ln \left (5\right )-\ln \left (\sec \left (\textit {\_Z} \right )^{2}\right )+2 \ln \left (-1+\tan \left (\textit {\_Z} \right )\right )+6 c_{1} -2 \textit {\_Z} \right )\right )-1\right ) {\mathrm e}^{-x^{2}} \]
✓ Solution by Mathematica
Time used: 7.286 (sec). Leaf size: 68
DSolve[y'[x] == (E^(2*x^2)*x*y[x]^3)/(1 + E^x^2*y[x]),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\log (y(x))-2 y(x)^2 \left (\frac {\log \left (e^{2 x^2} y(x)^2+2 e^{x^2} y(x)+2\right )}{4 y(x)^2}-\frac {\arctan \left (e^{x^2} y(x)+1\right )}{2 y(x)^2}\right )=c_1,y(x)\right ] \]