Internal problem ID [1957]
Book: Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath.
Boston. 1964
Section: Exercise 8, page 34
Problem number: 16.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_exact, [_Abel, `2nd type`, `class B`]]
\[ \boxed {y^{2} \csc \left (x \right )^{2}+6 y x -\left (2 y \cot \left (x \right )-3 x^{2}\right ) y^{\prime }=2} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 71
dsolve(y(x)^2*csc(x)^2+6*x*y(x)-2=(2*y(x)*cot(x)-3*x^2)*diff(y(x),x),y(x), singsol=all)
\begin{align*} y = \frac {3 \tan \left (x \right ) x^{2}}{2}-\frac {\sqrt {9 \tan \left (x \right )^{2} x^{4}+4 c_{1} \tan \left (x \right )-8 x \tan \left (x \right )}}{2} y = \frac {3 \tan \left (x \right ) x^{2}}{2}+\frac {\sqrt {9 \tan \left (x \right )^{2} x^{4}+4 c_{1} \tan \left (x \right )-8 x \tan \left (x \right )}}{2} \end{align*}
✓ Solution by Mathematica
Time used: 31.692 (sec). Leaf size: 201
DSolve[y[x]^2*Csc[x]^2+6*x*y[x]-2==(2*y[x]*Cot[x]-3*x^2)*y'[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {3}{2} x^2 \tan (x)-\frac {\csc (2 x) \sqrt {-\left (\tan (x) \left (16 \cos ^2(x) \arcsin \left (\sqrt {\sin ^2(x)}\right )-9 x^4 e^{\text {arctanh}(\cos (2 x))}+\cos (2 x) \left (9 x^4 e^{\text {arctanh}(\cos (2 x))}-4 c_1\right )-4 c_1\right )\right )}}{2 \sqrt {\csc (2 x) e^{\text {arctanh}(\cos (2 x))}}} y(x)\to \frac {3}{2} x^2 \tan (x)+\frac {\csc (2 x) \sqrt {-\left (\tan (x) \left (16 \cos ^2(x) \arcsin \left (\sqrt {\sin ^2(x)}\right )-9 x^4 e^{\text {arctanh}(\cos (2 x))}+\cos (2 x) \left (9 x^4 e^{\text {arctanh}(\cos (2 x))}-4 c_1\right )-4 c_1\right )\right )}}{2 \sqrt {\csc (2 x) e^{\text {arctanh}(\cos (2 x))}}} \end{align*}