Internal problem ID [8915]
Internal file name [OUTPUT/7850_Monday_June_06_2022_12_46_59_AM_45624977/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 581.
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(x)]`]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }-\frac {1+2 F \left (\frac {1+4 x^{2} y}{4 x^{2}}\right ) x}{2 x^{3}}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & -2 y^{\prime } x^{3}+2 F \left (\frac {1+4 x^{2} y}{4 x^{2}}\right ) x +1=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {-1-2 F \left (\frac {1+4 x^{2} y}{4 x^{2}}\right ) x}{2 x^{3}} \end {array} \]
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`[2*x^2, 1/x]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 50
dsolve(diff(y(x),x) = 1/2*(1+2*F(1/4*(4*x^2*y(x)+1)/x^2)*x)/x^3,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {4 \operatorname {RootOf}\left (F \left (\textit {\_Z} \right )\right ) x^{2}-1}{4 x^{2}} \\ y \left (x \right ) &= \frac {4 \operatorname {RootOf}\left (\left (\int _{}^{\textit {\_Z}}\frac {1}{F \left (\textit {\_a} \right )}d \textit {\_a} \right ) x +c_{1} x +1\right ) x^{2}-1}{4 x^{2}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.254 (sec). Leaf size: 144
DSolve[y'[x] == (1/2 + x*F[(1/4 + x^2*y[x])/x^2])/x^3,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}-\frac {F\left (\frac {K[2] x^2+\frac {1}{4}}{x^2}\right ) \int _1^x-\frac {F'\left (\frac {K[2] K[1]^2+\frac {1}{4}}{K[1]^2}\right )}{2 F\left (\frac {K[2] K[1]^2+\frac {1}{4}}{K[1]^2}\right )^2 K[1]^3}dK[1]+1}{F\left (\frac {K[2] x^2+\frac {1}{4}}{x^2}\right )}dK[2]+\int _1^x\left (\frac {1}{K[1]^2}+\frac {1}{2 K[1]^3 F\left (\frac {y(x) K[1]^2+\frac {1}{4}}{K[1]^2}\right )}\right )dK[1]=c_1,y(x)\right ] \]