Internal problem ID [8922]
Internal file name [OUTPUT/7857_Monday_June_06_2022_12_47_45_AM_42811900/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 588.
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.
\[ \boxed {y^{\prime }-\frac {x +F \left (-\left (-y+x \right ) \left (y+x \right )\right )}{y}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y y^{\prime }-F \left (\left (y-x \right ) \left (y+x \right )\right )-x =0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {x +F \left (\left (y-x \right ) \left (y+x \right )\right )}{y} \end {array} \]
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]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 67
dsolve(diff(y(x),x) = (x+F(-(x-y(x))*(x+y(x))))/y(x),y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \operatorname {RootOf}\left (F \left (\textit {\_Z}^{2}-x^{2}\right )\right ) \\ y \left (x \right ) &= \sqrt {x^{2}+\operatorname {RootOf}\left (-2 x +\int _{}^{\textit {\_Z}}\frac {1}{F \left (\textit {\_a} \right )}d \textit {\_a} +2 c_{1} \right )} \\ y \left (x \right ) &= -\sqrt {x^{2}+\operatorname {RootOf}\left (-2 x +\int _{}^{\textit {\_Z}}\frac {1}{F \left (\textit {\_a} \right )}d \textit {\_a} +2 c_{1} \right )} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.211 (sec). Leaf size: 109
DSolve[y'[x] == (x + F[(-x + y[x])*(x + y[x])])/y[x],y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}\left (-\frac {K[2]}{F((K[2]-x) (x+K[2]))}-\int _1^x-\frac {2 K[1] K[2] F'((K[2]-K[1]) (K[1]+K[2]))}{F((K[2]-K[1]) (K[1]+K[2]))^2}dK[1]\right )dK[2]+\int _1^x\left (\frac {K[1]}{F((y(x)-K[1]) (K[1]+y(x)))}+1\right )dK[1]=c_1,y(x)\right ] \]