Internal problem ID [8924]
Internal file name [OUTPUT/7859_Monday_June_06_2022_12_47_59_AM_23290139/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 590.
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.
\[ \boxed {y^{\prime }-\frac {x}{-y+F \left (x^{2}+y^{2}\right )}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y y^{\prime }-y^{\prime } F \left (x^{2}+y^{2}\right )+x =0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {x}{y-F \left (x^{2}+y^{2}\right )} \end {array} \]
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`[-y/x, 1]
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 40
dsolve(diff(y(x),x) = x/(-y(x)+F(y(x)^2+x^2)),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 )+\frac {\left (\int _{}^{y \left (x \right )^{2}+x^{2}}\frac {1}{F \left (\textit {\_a} \right )}d \textit {\_a} \right )}{2}-c_{1} &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.254 (sec). Leaf size: 94
DSolve[y'[x] == x/(F[x^2 + y[x]^2] - y[x]),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}\left (-\frac {K[2]}{F\left (x^2+K[2]^2\right )}-\int _1^x\frac {2 K[1] K[2] F'\left (K[1]^2+K[2]^2\right )}{F\left (K[1]^2+K[2]^2\right )^2}dK[1]+1\right )dK[2]+\int _1^x-\frac {K[1]}{F\left (K[1]^2+y(x)^2\right )}dK[1]=c_1,y(x)\right ] \]