Internal problem ID [8938]
Internal file name [OUTPUT/7873_Monday_June_06_2022_12_49_20_AM_82580329/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 604.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[`x=_G(y,y')`]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }-\frac {2 y^{3}}{1+2 F \left (\frac {1+4 y^{2} x}{y^{2}}\right ) y}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 2 y^{3}-2 y^{\prime } y F \left (\frac {1+4 y^{2} x}{y^{2}}\right )-y^{\prime }=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {2 y^{3}}{-2 F \left (\frac {1+4 y^{2} x}{y^{2}}\right ) y-1} \end {array} \]
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, y^2]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 47
dsolve(diff(y(x),x) = 2*y(x)^3/(1+2*F((1+4*x*y(x)^2)/y(x)^2)*y(x)),y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \operatorname {RootOf}\left (F \left (\frac {4 x \,\textit {\_Z}^{2}+1}{\textit {\_Z}^{2}}\right )\right ) \\ -c_{1} -\frac {1}{y \left (x \right )}-\frac {\left (\int _{}^{4 x +\frac {1}{y \left (x \right )^{2}}}\frac {1}{F \left (\textit {\_a} \right )}d \textit {\_a} \right )}{4} &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.398 (sec). Leaf size: 143
DSolve[y'[x] == (2*y[x]^3)/(1 + 2*F[(1 + 4*x*y[x]^2)/y[x]^2]*y[x]),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}\left (-\int _1^x\frac {\left (\frac {8 K[1]}{K[2]}-\frac {2 \left (4 K[1] K[2]^2+1\right )}{K[2]^3}\right ) F'\left (\frac {4 K[1] K[2]^2+1}{K[2]^2}\right )}{F\left (\frac {4 K[1] K[2]^2+1}{K[2]^2}\right )^2}dK[1]+\frac {1}{K[2]^2}+\frac {1}{2 F\left (\frac {4 x K[2]^2+1}{K[2]^2}\right ) K[2]^3}\right )dK[2]+\int _1^x-\frac {1}{F\left (\frac {4 K[1] y(x)^2+1}{y(x)^2}\right )}dK[1]=c_1,y(x)\right ] \]