Internal problem ID [8936]
Internal file name [OUTPUT/7871_Monday_June_06_2022_12_49_09_AM_40668385/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 602.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[NONE]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }-\frac {y^{2} \left (2+F \left (\frac {x^{2}-y}{y x^{2}}\right ) x^{2}\right )}{x^{3}}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{2} F \left (-\frac {-x^{2}+y}{y x^{2}}\right ) x^{2}-y^{\prime } x^{3}+2 y^{2}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {-y^{2} F \left (-\frac {-x^{2}+y}{y x^{2}}\right ) x^{2}-2 y^{2}}{x^{3}} \end {array} \]
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*x, 1/x^2*y^2]
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 51
dsolve(diff(y(x),x) = 1/x^3*y(x)^2*(2+F((x^2-y(x))/y(x)/x^2)*x^2),y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {x^{2}}{\operatorname {RootOf}\left (F \left (\textit {\_Z} \right )\right ) x^{2}+1} \\ y \left (x \right ) &= \frac {x^{2}}{\operatorname {RootOf}\left (-\ln \left (x \right )-\left (\int _{}^{\textit {\_Z}}\frac {1}{F \left (\textit {\_a} \right )}d \textit {\_a} \right )+c_{1} \right ) x^{2}+1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.43 (sec). Leaf size: 167
DSolve[y'[x] == ((2 + x^2*F[(x^2 - y[x])/(x^2*y[x])])*y[x]^2)/x^3,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}\left (-\int _1^x-\frac {2 \left (-\frac {K[1]^2-K[2]}{K[1]^2 K[2]^2}-\frac {1}{K[1]^2 K[2]}\right ) F'\left (\frac {K[1]^2-K[2]}{K[1]^2 K[2]}\right )}{F\left (\frac {K[1]^2-K[2]}{K[1]^2 K[2]}\right )^2 K[1]^3}dK[1]-\frac {1}{F\left (\frac {x^2-K[2]}{x^2 K[2]}\right ) K[2]^2}\right )dK[2]+\int _1^x\left (\frac {1}{K[1]}+\frac {2}{K[1]^3 F\left (\frac {K[1]^2-y(x)}{K[1]^2 y(x)}\right )}\right )dK[1]=c_1,y(x)\right ] \]