32.6.19 problem Exercise 12.19, page 103

Internal problem ID [5884]
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
Date solved : Tuesday, January 28, 2025 at 03:08:40 PM
CAS classification : [NONE]

\begin{align*} \left (x y \sqrt {x^{2}-y^{2}}+x \right ) y^{\prime }&=y-x^{2} \sqrt {x^{2}-y^{2}} \end{align*}

✓ Solution by Maple

Time used: 0.040 (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: 1.757 (sec). Leaf size: 44

DSolve[(x*y[x]*Sqrt[x^2-y[x]^2]+x)*D[y[x],x]==y[x]-x^2*Sqrt[x^2-y[x]^2],y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [-\arctan \left (\frac {\sqrt {x^2-y(x)^2}}{y(x)}\right )+\frac {x^2}{2}+\frac {y(x)^2}{2}=c_1,y(x)\right ] \]