Internal problem ID [3976]
Internal file name [OUTPUT/3469_Sunday_June_05_2022_09_22_27_AM_55036887/index.tex]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 25
Problem number: 730.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[`y=_G(x,y')`]
Unable to solve or complete the solution.
\[ \boxed {x \left (1-\sqrt {x^{2}-y^{2}}\right ) y^{\prime }-y=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x \left (1-\sqrt {x^{2}-y^{2}}\right ) y^{\prime }-y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {y}{x \left (1-\sqrt {x^{2}-y^{2}}\right )} \end {array} \]
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 = 5`[0, (x^2-y^2)^(1/2)/(-1+(x^2-y^2)^(1/2))]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 27
dsolve(x*(1-sqrt(x^2-y(x)^2))*diff(y(x),x) = y(x),y(x), singsol=all)
\[ y \left (x \right )-\arctan \left (\frac {y \left (x \right )}{\sqrt {x^{2}-y \left (x \right )^{2}}}\right )-c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.84 (sec). Leaf size: 29
DSolve[x(1-Sqrt[x^2-y[x]^2])y'[x]==y[x],y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\arctan \left (\frac {\sqrt {x^2-y(x)^2}}{y(x)}\right )+y(x)=c_1,y(x)\right ] \]