Internal problem ID [4540]
Internal file name [OUTPUT/4033_Sunday_June_05_2022_12_12_27_PM_21590560/index.tex]
Book: Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section: Chapter 2. Special types of differential equations of the first kind. Lesson 12,
Miscellaneous Methods
Problem number: Exercise 12.19, page 103.
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.
\[ \boxed {\left (x y \sqrt {x^{2}-y^{2}}+x \right ) y^{\prime }-y+x^{2} \sqrt {x^{2}-y^{2}}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (x y \sqrt {x^{2}-y^{2}}+x \right ) y^{\prime }-y+x^{2} \sqrt {x^{2}-y^{2}}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {y-x^{2} \sqrt {x^{2}-y^{2}}}{x y \sqrt {x^{2}-y^{2}}+x} \end {array} \]
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: 0.031 (sec). Leaf size: 34
dsolve((x*y(x)*sqrt(x^2-y(x)^2)+x)*diff(y(x),x)=y(x)-x^2*sqrt(x^2-y(x)^2),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: 1.772 (sec). Leaf size: 44
DSolve[(x*y[x]*Sqrt[x^2-y[x]^2]+x)*y'[x]==y[x]-x^2*Sqrt[x^2-y[x]^2],y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [-\arctan \left (\frac {\sqrt {x^2-y(x)^2}}{y(x)}\right )+\frac {x^2}{2}+\frac {y(x)^2}{2}=c_1,y(x)\right ] \]