Internal problem ID [9801]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second
edition
Section: Chapter 1, section 1.2. Riccati Equation. subsection 1.2.6-4. Equations with cotangent.
Problem number: 46.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {\left (a \cot \left (\lambda x \right )+b \right ) y^{\prime }-y^{2}-c \cot \left (\mu x \right ) y+d^{2}-c d \cot \left (\mu x \right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.017 (sec). Leaf size: 251
dsolve((a*cot(lambda*x)+b)*diff(y(x),x)=y(x)^2+c*cot(mu*x)*y(x)-d^2+c*d*cot(mu*x),y(x), singsol=all)
\[ y \relax (x ) = -d -\frac {{\mathrm e}^{\int \frac {c \cot \left (\mu x \right )}{a \cot \left (\lambda x \right )+b}d x} \left (a \cot \left (\lambda x \right )+b \right )^{\frac {2 a d}{\lambda \left (a^{2}+b^{2}\right )}} \left (\cot ^{2}\left (\lambda x \right )+1\right )^{-\frac {a d}{\lambda \left (a^{2}+b^{2}\right )}} {\mathrm e}^{\frac {\pi b d}{\lambda \left (a^{2}+b^{2}\right )}} {\mathrm e}^{-\frac {2 d b \,\mathrm {arccot}\left (\cot \left (\lambda x \right )\right )}{\lambda \left (a^{2}+b^{2}\right )}}}{\int \frac {{\mathrm e}^{\int \frac {c \cot \left (\mu x \right )}{a \cot \left (\lambda x \right )+b}d x} \left (a \cot \left (\lambda x \right )+b \right )^{\frac {2 a d}{\lambda \left (a^{2}+b^{2}\right )}} \left (\cot ^{2}\left (\lambda x \right )+1\right )^{-\frac {a d}{\lambda \left (a^{2}+b^{2}\right )}} {\mathrm e}^{\frac {\pi b d}{\lambda \left (a^{2}+b^{2}\right )}} {\mathrm e}^{-\frac {2 d b \,\mathrm {arccot}\left (\cot \left (\lambda x \right )\right )}{\lambda \left (a^{2}+b^{2}\right )}}}{a \cot \left (\lambda x \right )+b}d x -c_{1}} \]
✓ Solution by Mathematica
Time used: 33.037 (sec). Leaf size: 799
DSolve[(a*Cot[\[Lambda]*x]+b)*y'[x]==y[x]^2+c*Cot[\[Mu]*x]*y[x]-d^2+c*d*Cot[\[Mu]*x],y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^x\frac {e^{-\int _1^{K[6]}-\frac {\csc (\mu K[5]) (-2 d \cos (\lambda K[5]-\mu K[5])+2 d \cos (\lambda K[5]+\mu K[5])+c \sin (\lambda K[5]-\mu K[5])+c \sin (\lambda K[5]+\mu K[5]))}{2 (a \cos (\lambda K[5])+b \sin (\lambda K[5]))}dK[5]} (-d \cos (\lambda K[6]-\mu K[6])+y(x) \cos (\lambda K[6]-\mu K[6])+d \cos (\lambda K[6]+\mu K[6])+c \sin (\lambda K[6]-\mu K[6])+c \sin (\lambda K[6]+\mu K[6])-\cos (\lambda K[6]+\mu K[6]) y(x))}{c \mu (b \cos (\lambda K[6]-\mu K[6])-b \cos (\lambda K[6]+\mu K[6])-a \sin (\lambda K[6]-\mu K[6])+a \sin (\lambda K[6]+\mu K[6])) (d+y(x))}dK[6]+\int _1^{y(x)}\left (-\int _1^x\left (\frac {e^{-\int _1^{K[6]}-\frac {\csc (\mu K[5]) (-2 d \cos (\lambda K[5]-\mu K[5])+2 d \cos (\lambda K[5]+\mu K[5])+c \sin (\lambda K[5]-\mu K[5])+c \sin (\lambda K[5]+\mu K[5]))}{2 (a \cos (\lambda K[5])+b \sin (\lambda K[5]))}dK[5]} (\cos (\lambda K[6]-\mu K[6])-\cos (\lambda K[6]+\mu K[6]))}{c \mu (d+K[7]) (b \cos (\lambda K[6]-\mu K[6])-b \cos (\lambda K[6]+\mu K[6])-a \sin (\lambda K[6]-\mu K[6])+a \sin (\lambda K[6]+\mu K[6]))}-\frac {e^{-\int _1^{K[6]}-\frac {\csc (\mu K[5]) (-2 d \cos (\lambda K[5]-\mu K[5])+2 d \cos (\lambda K[5]+\mu K[5])+c \sin (\lambda K[5]-\mu K[5])+c \sin (\lambda K[5]+\mu K[5]))}{2 (a \cos (\lambda K[5])+b \sin (\lambda K[5]))}dK[5]} (-d \cos (\lambda K[6]-\mu K[6])+K[7] \cos (\lambda K[6]-\mu K[6])+d \cos (\lambda K[6]+\mu K[6])-\cos (\lambda K[6]+\mu K[6]) K[7]+c \sin (\lambda K[6]-\mu K[6])+c \sin (\lambda K[6]+\mu K[6]))}{c \mu (d+K[7])^2 (b \cos (\lambda K[6]-\mu K[6])-b \cos (\lambda K[6]+\mu K[6])-a \sin (\lambda K[6]-\mu K[6])+a \sin (\lambda K[6]+\mu K[6]))}\right )dK[6]-\frac {e^{-\int _1^x-\frac {\csc (\mu K[5]) (-2 d \cos (\lambda K[5]-\mu K[5])+2 d \cos (\lambda K[5]+\mu K[5])+c \sin (\lambda K[5]-\mu K[5])+c \sin (\lambda K[5]+\mu K[5]))}{2 (a \cos (\lambda K[5])+b \sin (\lambda K[5]))}dK[5]}}{c \mu (d+K[7])^2}\right )dK[7]=c_1,y(x)\right ] \]