Internal problem ID [11700]
Book: Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi.
2004.
Section: Chapter 2, Section 2.4. Special integrating factors and transformations. Exercises page
67
Problem number: 2.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, `_with_symmetry_[F(x),G(x)]`]]
\[ \boxed {\tan \left (y\right )+\left (x -x^{2} \tan \left (y\right )\right ) y^{\prime }=-2 x} \]
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 134
dsolve((2*x+tan(y(x)))+(x-x^2*tan(y(x)))*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \arctan \left (\frac {-\sqrt {x^{4}-c_{1}^{2}+x^{2}}\, x -c_{1}}{\left (x^{2}+1\right ) x}, \frac {-c_{1} x +\sqrt {x^{4}-c_{1}^{2}+x^{2}}}{\left (x^{2}+1\right ) x}\right ) \\ y \left (x \right ) &= \arctan \left (\frac {\sqrt {x^{4}-c_{1}^{2}+x^{2}}\, x -c_{1}}{\left (x^{2}+1\right ) x}, \frac {-c_{1} x -\sqrt {x^{4}-c_{1}^{2}+x^{2}}}{\left (x^{2}+1\right ) x}\right ) \\ \end{align*}
✓ Solution by Mathematica
Time used: 38.283 (sec). Leaf size: 177
DSolve[(2*x+Tan[y[x]])+(x-x^2*Tan[y[x]])*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\arccos \left (-\frac {c_1 x^2+\sqrt {x^6+x^4-c_1{}^2 x^2}}{x^4+x^2}\right ) \\ y(x)\to \arccos \left (-\frac {c_1 x^2+\sqrt {x^6+x^4-c_1{}^2 x^2}}{x^4+x^2}\right ) \\ y(x)\to -\arccos \left (\frac {\sqrt {x^6+x^4-c_1{}^2 x^2}-c_1 x^2}{x^4+x^2}\right ) \\ y(x)\to \arccos \left (\frac {\sqrt {x^6+x^4-c_1{}^2 x^2}-c_1 x^2}{x^4+x^2}\right ) \\ \end{align*}