32.4.17 problem Recognizable Exact Differential equations. Integrating factors. Exercise 10.9, page 90

Internal problem ID [5828]
Book : Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section : Chapter 2. Special types of differential equations of the first kind. Lesson 10
Problem number : Recognizable Exact Differential equations. Integrating factors. Exercise 10.9, page 90
Date solved : Monday, January 27, 2025 at 01:20:09 PM
CAS classification : [_exact]

\begin{align*} \arctan \left (y x \right )+\frac {y x -2 x y^{2}}{1+x^{2} y^{2}}+\frac {\left (x^{2}-2 x^{2} y\right ) y^{\prime }}{1+x^{2} y^{2}}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.201 (sec). Leaf size: 22

dsolve((arctan(x*y(x))+(x*y(x)-2*x*y(x)^2)/(1+x^2*y(x)^2))+((x^2-2*x^2*y(x))/(1+x^2*y(x)^2))*diff(y(x),x)=0,y(x), singsol=all)
 

\[ y \left (x \right ) = \frac {\tan \left (\operatorname {RootOf}\left (x \textit {\_Z} -\ln \left (\sec \left (\textit {\_Z} \right )^{2}\right )+c_{1} \right )\right )}{x} \]

✓ Solution by Mathematica

Time used: 0.205 (sec). Leaf size: 26

DSolve[(ArcTan[x*y[x]]+(x*y[x]-2*x*y[x]^2)/(1+x^2*y[x]^2))+((x^2-2*x^2*y[x])/(1+x^2*y[x]^2))*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\log \left (x^2 y(x)^2+1\right )-x \arctan (x y(x))=c_1,y(x)\right ] \]