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]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {y^{2}+y \tan \relax (x )+\tan ^{2}\relax (x )}{\sin \relax (x )^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 115
dsolve(diff(y(x),x)=(y(x)^2+y(x)*tan(x)+tan(x)^2)/sin(x)^2,y(x), singsol=all)
\[ y \relax (x ) = -\frac {\cos \relax (x ) \left (-c_{1} \sin \left (\frac {\ln \left (\sin \relax (x )-1\right )}{2}-\ln \left (\sin \relax (x )\right )+\frac {\ln \left (\sin \relax (x )+1\right )}{2}\right )+\cos \left (\frac {\ln \left (\sin \relax (x )-1\right )}{2}-\ln \left (\sin \relax (x )\right )+\frac {\ln \left (\sin \relax (x )+1\right )}{2}\right )\right ) \sin \relax (x )}{\left (\sin \relax (x )-1\right ) \left (\sin \relax (x )+1\right ) \left (c_{1} \cos \left (\frac {\ln \left (\sin \relax (x )-1\right )}{2}-\ln \left (\sin \relax (x )\right )+\frac {\ln \left (\sin \relax (x )+1\right )}{2}\right )+\sin \left (\frac {\ln \left (\sin \relax (x )-1\right )}{2}-\ln \left (\sin \relax (x )\right )+\frac {\ln \left (\sin \relax (x )+1\right )}{2}\right )\right )} \]
✓ Solution by Mathematica
Time used: 0.709 (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]
\begin{align*} y(x)\to \tan (x) \tan (\log (\sin (x))-\log (\cos (x))+c_1) \\ \end{align*}