Internal problem ID [1023]
Book: Elementary differential equations with boundary value problems. William F. Trench.
Brooks/Cole 2001
Section: Chapter 2, First order equations. Transformation of Nonlinear Equations into Separable
Equations. Section 2.4 Page 68
Problem number: 48.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }-\frac {y^{2}+y \tan \left (x \right )+\tan \left (x \right )^{2}}{\sin \left (x \right )^{2}}=0} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 102
dsolve(diff(y(x),x)=(y(x)^2+y(x)*tan(x)+tan(x)^2)/sin(x)^2,y(x), singsol=all)
\[ y \left (x \right ) = -\frac {\tan \left (x \right ) \left (c_{1} \sin \left (\frac {\ln \left (\sin \left (x \right )-1\right )}{2}+\frac {\ln \left (\sin \left (x \right )+1\right )}{2}-\ln \left (\sin \left (x \right )\right )\right )-\cos \left (\frac {\ln \left (\sin \left (x \right )-1\right )}{2}+\frac {\ln \left (\sin \left (x \right )+1\right )}{2}-\ln \left (\sin \left (x \right )\right )\right )\right )}{c_{1} \cos \left (\frac {\ln \left (\sin \left (x \right )-1\right )}{2}+\frac {\ln \left (\sin \left (x \right )+1\right )}{2}-\ln \left (\sin \left (x \right )\right )\right )+\sin \left (\frac {\ln \left (\sin \left (x \right )-1\right )}{2}+\frac {\ln \left (\sin \left (x \right )+1\right )}{2}-\ln \left (\sin \left (x \right )\right )\right )} \]
✓ Solution by Mathematica
Time used: 0.678 (sec). Leaf size: 20
DSolve[y'[x]==(y[x]^2+y[x]*Tan[x]+Tan[x]^2)/Sin[x]^2,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \tan (x) \tan (\log (\sin (x))-\log (\cos (x))+c_1) \]