Internal problem ID [8399]
Internal file name [OUTPUT/7332_Sunday_June_05_2022_05_50_56_PM_78192277/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 62.
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.
\[ \boxed {y^{\prime }-\frac {y-x^{2} \sqrt {x^{2}-y^{2}}}{x y \sqrt {x^{2}-y^{2}}+x}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y \sqrt {-\left (y-x \right ) \left (y+x \right )}\, y^{\prime } x +\sqrt {-\left (y-x \right ) \left (y+x \right )}\, x^{2}+x y^{\prime }-y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-\sqrt {-\left (y-x \right ) \left (y+x \right )}\, x^{2}+y}{y \sqrt {-\left (y-x \right ) \left (y+x \right )}\, x +x} \end {array} \]
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 = 5`[0, 1/((x^2-y^2)^(1/2)*y+1)*(x^2-y^2)^(1/2)]
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 34
dsolve(diff(y(x),x) - (y(x)-x^2*sqrt(x^2-y(x)^2))/(x*y(x)*sqrt(x^2-y(x)^2)+x)=0,y(x), singsol=all)
\[ \frac {y \left (x \right )^{2}}{2}+\arctan \left (\frac {y \left (x \right )}{\sqrt {x^{2}-y \left (x \right )^{2}}}\right )+\frac {x^{2}}{2}-c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 1.836 (sec). Leaf size: 44
DSolve[y'[x] - (y[x]-x^2*Sqrt[x^2-y[x]^2])/(x*y[x]*Sqrt[x^2-y[x]^2]+x)==0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [-\arctan \left (\frac {\sqrt {x^2-y(x)^2}}{y(x)}\right )+\frac {x^2}{2}+\frac {y(x)^2}{2}=c_1,y(x)\right ] \]