Internal problem ID [8971]
Internal file name [OUTPUT/7906_Monday_June_06_2022_12_53_23_AM_13760177/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 637.
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.
\[ \boxed {y^{\prime }-\frac {{\mathrm e}^{-x^{2}} x}{y \,{\mathrm e}^{x^{2}}+1}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime } y \,{\mathrm e}^{x^{2}}-{\mathrm e}^{-x^{2}} x +y^{\prime }=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {{\mathrm e}^{-x^{2}} x}{y \,{\mathrm e}^{x^{2}}+1} \end {array} \]
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: 0.0 (sec). Leaf size: 84
dsolve(diff(y(x),x) = 1/(y(x)*exp(x^2)+1)*exp(-x^2)*x,y(x), singsol=all)
\[ y \left (x \right ) = -\frac {\tan \left (\operatorname {RootOf}\left (2 x^{2}-\ln \left (2\right )+\ln \left (5\right )-\ln \left (\sec \left (\textit {\_Z} \right )^{2}\right )+2 \ln \left (-1+\tan \left (\textit {\_Z} \right )\right )+6 c_{1} -2 \textit {\_Z} \right )\right ) {\mathrm e}^{-x^{2}}}{\tan \left (\operatorname {RootOf}\left (2 x^{2}-\ln \left (2\right )+\ln \left (5\right )-\ln \left (\sec \left (\textit {\_Z} \right )^{2}\right )+2 \ln \left (-1+\tan \left (\textit {\_Z} \right )\right )+6 c_{1} -2 \textit {\_Z} \right )\right )-1} \]
✓ Solution by Mathematica
Time used: 7.089 (sec). Leaf size: 62
DSolve[y'[x] == x/(E^x^2*(1 + E^x^2*y[x])),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [-\frac {1}{2} \arctan \left (2 e^{x^2} y(x)+1\right )-\frac {1}{4} \log \left (2 e^{2 x^2} y(x)^2+2 e^{x^2} y(x)+1\right )+\frac {1}{2} \log \left (e^{x^2}\right )=c_1,y(x)\right ] \]