Internal problem ID [9130]
Internal file name [OUTPUT/8065_Monday_June_06_2022_01_37_58_AM_11695995/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 796.
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`]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }-\frac {y^{3} x \,{\mathrm e}^{3 x^{2}} {\mathrm e}^{-\frac {9 x^{2}}{2}}}{3 \left (3 \,{\mathrm e}^{\frac {3 x^{2}}{2}}+{\mathrm e}^{\frac {3 x^{2}}{2}} y+3 y\right )}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & -y^{3} x \,{\mathrm e}^{3 x^{2}}+3 y^{\prime } y \,{\mathrm e}^{\frac {3 x^{2}}{2}} {\mathrm e}^{\frac {9 x^{2}}{2}}+9 y^{\prime } y \,{\mathrm e}^{\frac {9 x^{2}}{2}}+9 y^{\prime } {\mathrm e}^{\frac {3 x^{2}}{2}} {\mathrm e}^{\frac {9 x^{2}}{2}}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {y^{3} x \,{\mathrm e}^{3 x^{2}}}{3 \,{\mathrm e}^{\frac {3 x^{2}}{2}} y \,{\mathrm e}^{\frac {9 x^{2}}{2}}+9 y \,{\mathrm e}^{\frac {9 x^{2}}{2}}+9 \,{\mathrm e}^{\frac {9 x^{2}}{2}} {\mathrm e}^{\frac {3 x^{2}}{2}}} \end {array} \]
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: 135
dsolve(diff(y(x),x) = 1/3*y(x)^3*x*exp(3*x^2)/(3*exp(3/2*x^2)+exp(3/2*x^2)*y(x)+3*y(x))/exp(9/2*x^2),y(x), singsol=all)
\[ y \left (x \right ) = \operatorname {RootOf}\left (81+\left (9 \,{\mathrm e}^{3 x^{2}}+27 \,{\mathrm e}^{\frac {3 x^{2}}{2}}+7 \,{\mathrm e}^{3 x^{2}+\operatorname {RootOf}\left (\left (42 \sinh \left (\frac {\left (c_{1} -5 \textit {\_Z} \right ) \sqrt {93}}{90}\right ) \sqrt {93}\, {\mathrm e}^{3 x^{2}+\textit {\_Z}} \cosh \left (\frac {\left (c_{1} -5 \textit {\_Z} \right ) \sqrt {93}}{90}\right )+406 \,{\mathrm e}^{3 x^{2}+\textit {\_Z}} \cosh \left (\frac {\left (c_{1} -5 \textit {\_Z} \right ) \sqrt {93}}{90}\right )^{2}-217 \,{\mathrm e}^{3 x^{2}+\textit {\_Z}}+93\right ) {\mathrm e}^{3 x^{2}}\right )}-3\right ) \textit {\_Z}^{2}+\left (54 \,{\mathrm e}^{\frac {3 x^{2}}{2}}+81\right ) \textit {\_Z} \right ) {\mathrm e}^{\frac {3 x^{2}}{2}} \]
✓ Solution by Mathematica
Time used: 7.509 (sec). Leaf size: 109
DSolve[y'[x] == (x*y[x]^3)/(3*E^((3*x^2)/2)*(3*E^((3*x^2)/2) + 3*y[x] + E^((3*x^2)/2)*y[x])),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {1}{62} \left (6 \sqrt {93} \text {arctanh}\left (\frac {\sqrt {\frac {3}{31}} \left (2 e^{\frac {3 x^2}{2}} (y(x)+3)+3 y(x)\right )}{y(x)}\right )-31 \log \left (9 e^{\frac {3 x^2}{2}} (y(x)+3) y(x)+3 e^{3 x^2} (y(x)+3)^2-y(x)^2\right )+62 \log \left (e^{\frac {3 x^2}{2}}\right )\right )+\log (y(x))=c_1,y(x)\right ] \]