Internal problem ID [8941]
Internal file name [OUTPUT/7876_Monday_June_06_2022_12_49_37_AM_73080633/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 607.
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.
\[ \boxed {y^{\prime }-\frac {2 y+F \left (\frac {y}{x^{2}}\right ) x^{3}}{x}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & F \left (\frac {y}{x^{2}}\right ) x^{3}-y^{\prime } x +2 y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {-2 y-F \left (\frac {y}{x^{2}}\right ) x^{3}}{x} \end {array} \]
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, 2*y/x]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 32
dsolve(diff(y(x),x) = (2*y(x)+F(1/x^2*y(x))*x^3)/x,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \operatorname {RootOf}\left (F \left (\textit {\_Z} \right )\right ) x^{2} \\ y \left (x \right ) &= \operatorname {RootOf}\left (-x +\int _{}^{\textit {\_Z}}\frac {1}{F \left (\textit {\_a} \right )}d \textit {\_a} +c_{1} \right ) x^{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.237 (sec). Leaf size: 121
DSolve[y'[x] == (x^3*F[y[x]/x^2] + 2*y[x])/x,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}-\frac {F\left (\frac {K[2]}{x^2}\right ) \int _1^x\left (\frac {2}{F\left (\frac {K[2]}{K[1]^2}\right ) K[1]^3}-\frac {2 K[2] F'\left (\frac {K[2]}{K[1]^2}\right )}{F\left (\frac {K[2]}{K[1]^2}\right )^2 K[1]^5}\right )dK[1] x^2+1}{x^2 F\left (\frac {K[2]}{x^2}\right )}dK[2]+\int _1^x\left (\frac {2 y(x)}{F\left (\frac {y(x)}{K[1]^2}\right ) K[1]^3}+1\right )dK[1]=c_1,y(x)\right ] \]