Internal problem ID [8951]
Internal file name [OUTPUT/7886_Monday_June_06_2022_12_50_34_AM_42025602/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 617.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_1st_order, `_with_symmetry_[F(x),G(y)]`]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }-\frac {F \left (\frac {\left (3+y\right ) {\mathrm e}^{\frac {3 x^{2}}{2}}}{3 y}\right ) x y^{2} {\mathrm e}^{3 x^{2}} {\mathrm e}^{-\frac {9 x^{2}}{2}}}{9}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & F \left (\frac {\left (3+y\right ) {\mathrm e}^{\frac {3 x^{2}}{2}}}{3 y}\right ) x y^{2} {\mathrm e}^{3 x^{2}}-9 y^{\prime } {\mathrm e}^{\frac {9 x^{2}}{2}}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {F \left (\frac {\left (3+y\right ) {\mathrm e}^{\frac {3 x^{2}}{2}}}{3 y}\right ) x y^{2} {\mathrm e}^{3 x^{2}}}{9 \,{\mathrm e}^{\frac {9 x^{2}}{2}}} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying homogeneous types: differential order: 1; looking for linear symmetries trying exact Looking for potential symmetries trying an equivalence to an Abel ODE trying 1st order ODE linearizable_by_differentiation --- Trying Lie symmetry methods, 1st order --- `, `-> Computing symmetries using: way = 3 `, `-> Computing symmetries using: way = 4`[1/x, y^2+3*y]
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 92
dsolve(diff(y(x),x) = 1/9*F(1/3*(3+y(x))*exp(3/2*x^2)/y(x))*x*y(x)^2*exp(3*x^2)/exp(9/2*x^2),y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \operatorname {RootOf}\left (F \left (\frac {\left (\textit {\_Z} +3\right ) {\mathrm e}^{\frac {3 x^{2}}{2}}}{3 \textit {\_Z}}\right ) \textit {\_Z} \,{\mathrm e}^{3 x^{2}}-9 \,{\mathrm e}^{\frac {9 x^{2}}{2}} \textit {\_Z} -27 \,{\mathrm e}^{\frac {9 x^{2}}{2}}\right ) \\ y \left (x \right ) &= -\frac {3 \,{\mathrm e}^{\frac {3 x^{2}}{2}}}{{\mathrm e}^{\frac {3 x^{2}}{2}}-3 \operatorname {RootOf}\left (-x^{2}-18 \left (\int _{}^{\textit {\_Z}}\frac {1}{F \left (\textit {\_a} \right )-27 \textit {\_a}}d \textit {\_a} \right )+2 c_{1} \right )} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.892 (sec). Leaf size: 615
DSolve[y'[x] == (x*F[(E^((3*x^2)/2)*(3 + y[x]))/(3*y[x])]*y[x]^2)/(9*E^((3*x^2)/2)),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}\left (\frac {9 e^{\frac {3 x^2}{2}}-F\left (\frac {e^{\frac {3 x^2}{2}} (K[2]+3)}{3 K[2]}\right )}{3 \left (\left (9 e^{\frac {3 x^2}{2}}-F\left (\frac {e^{\frac {3 x^2}{2}} (K[2]+3)}{3 K[2]}\right )\right ) K[2]+27 e^{\frac {3 x^2}{2}}\right )}-\int _1^x\left (-\frac {K[2] \left (\frac {e^{\frac {3 K[1]^2}{2}}}{3 K[2]}-\frac {e^{\frac {3 K[1]^2}{2}} (K[2]+3)}{3 K[2]^2}\right ) F'\left (\frac {e^{\frac {3 K[1]^2}{2}} (K[2]+3)}{3 K[2]}\right ) K[1]}{-9 e^{\frac {3 K[1]^2}{2}} K[2]+F\left (\frac {e^{\frac {3 K[1]^2}{2}} (K[2]+3)}{3 K[2]}\right ) K[2]-27 e^{\frac {3 K[1]^2}{2}}}+\frac {F\left (\frac {e^{\frac {3 K[1]^2}{2}} (K[2]+3)}{3 K[2]}\right ) K[2] \left (F\left (\frac {e^{\frac {3 K[1]^2}{2}} (K[2]+3)}{3 K[2]}\right )-9 e^{\frac {3 K[1]^2}{2}}+K[2] \left (\frac {e^{\frac {3 K[1]^2}{2}}}{3 K[2]}-\frac {e^{\frac {3 K[1]^2}{2}} (K[2]+3)}{3 K[2]^2}\right ) F'\left (\frac {e^{\frac {3 K[1]^2}{2}} (K[2]+3)}{3 K[2]}\right )\right ) K[1]}{\left (-9 e^{\frac {3 K[1]^2}{2}} K[2]+F\left (\frac {e^{\frac {3 K[1]^2}{2}} (K[2]+3)}{3 K[2]}\right ) K[2]-27 e^{\frac {3 K[1]^2}{2}}\right )^2}-\frac {F\left (\frac {e^{\frac {3 K[1]^2}{2}} (K[2]+3)}{3 K[2]}\right ) K[1]}{-9 e^{\frac {3 K[1]^2}{2}} K[2]+F\left (\frac {e^{\frac {3 K[1]^2}{2}} (K[2]+3)}{3 K[2]}\right ) K[2]-27 e^{\frac {3 K[1]^2}{2}}}\right )dK[1]-\frac {1}{3 K[2]}\right )dK[2]+\int _1^x-\frac {F\left (\frac {e^{\frac {3 K[1]^2}{2}} (y(x)+3)}{3 y(x)}\right ) K[1] y(x)}{-9 e^{\frac {3 K[1]^2}{2}} y(x)+F\left (\frac {e^{\frac {3 K[1]^2}{2}} (y(x)+3)}{3 y(x)}\right ) y(x)-27 e^{\frac {3 K[1]^2}{2}}}dK[1]=c_1,y(x)\right ] \]