Internal problem ID [10544]
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]
\[ \boxed {\left (a \cot \left (\lambda x \right )+b \right ) y^{\prime }-y^{2}-c \cot \left (x \mu \right ) y=c d \cot \left (x \mu \right )-d^{2}} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 366
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)
\[ y \left (x \right ) = \frac {-{\mathrm e}^{\frac {\lambda c \left (a^{2}+b^{2}\right ) \left (\int \frac {\cot \left (x \mu \right )}{a \cot \left (x \lambda \right )+b}d x \right )-2 d \left (\operatorname {arccot}\left (\cot \left (x \lambda \right )\right )-\frac {\pi }{2}\right ) b}{\lambda \left (a^{2}+b^{2}\right )}} \left (\csc \left (x \lambda \right )^{2}\right )^{-\frac {a d}{\lambda \left (a^{2}+b^{2}\right )}} \left (a \cot \left (x \lambda \right )+b \right )^{\frac {2 a d}{\lambda \left (a^{2}+b^{2}\right )}}-d \left (\int \left (a \cot \left (x \lambda \right )+b \right )^{\frac {\left (-a^{2}-b^{2}\right ) \lambda +2 a d}{\lambda \left (a^{2}+b^{2}\right )}} {\mathrm e}^{\frac {\lambda c \left (a^{2}+b^{2}\right ) \left (\int \frac {\cot \left (x \mu \right )}{a \cot \left (x \lambda \right )+b}d x \right )-2 d \left (\operatorname {arccot}\left (\cot \left (x \lambda \right )\right )-\frac {\pi }{2}\right ) b}{\lambda \left (a^{2}+b^{2}\right )}} \left (\csc \left (x \lambda \right )^{2}\right )^{-\frac {a d}{\lambda \left (a^{2}+b^{2}\right )}}d x -c_{1} \right )}{\int \left (a \cot \left (x \lambda \right )+b \right )^{\frac {\left (-a^{2}-b^{2}\right ) \lambda +2 a d}{\lambda \left (a^{2}+b^{2}\right )}} {\mathrm e}^{\frac {\lambda c \left (a^{2}+b^{2}\right ) \left (\int \frac {\cot \left (x \mu \right )}{a \cot \left (x \lambda \right )+b}d x \right )-2 d \left (\operatorname {arccot}\left (\cot \left (x \lambda \right )\right )-\frac {\pi }{2}\right ) b}{\lambda \left (a^{2}+b^{2}\right )}} \left (\csc \left (x \lambda \right )^{2}\right )^{-\frac {a d}{\lambda \left (a^{2}+b^{2}\right )}}d x -c_{1}} \]
✓ Solution by Mathematica
Time used: 87.594 (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]
\[ \text {Solve}\left [\int _1^x\frac {e^{-\int _1^{K[2]}-\frac {\csc (\mu K[1]) (-2 d \cos (\lambda K[1]-\mu K[1])+2 d \cos (\lambda K[1]+\mu K[1])+c \sin (\lambda K[1]-\mu K[1])+c \sin (\lambda K[1]+\mu K[1]))}{2 (a \cos (\lambda K[1])+b \sin (\lambda K[1]))}dK[1]} (-d \cos (\lambda K[2]-\mu K[2])+y(x) \cos (\lambda K[2]-\mu K[2])+d \cos (\lambda K[2]+\mu K[2])+c \sin (\lambda K[2]-\mu K[2])+c \sin (\lambda K[2]+\mu K[2])-\cos (\lambda K[2]+\mu K[2]) y(x))}{c \mu (b \cos (\lambda K[2]-\mu K[2])-b \cos (\lambda K[2]+\mu K[2])-a \sin (\lambda K[2]-\mu K[2])+a \sin (\lambda K[2]+\mu K[2])) (d+y(x))}dK[2]+\int _1^{y(x)}\left (-\int _1^x\left (\frac {e^{-\int _1^{K[2]}-\frac {\csc (\mu K[1]) (-2 d \cos (\lambda K[1]-\mu K[1])+2 d \cos (\lambda K[1]+\mu K[1])+c \sin (\lambda K[1]-\mu K[1])+c \sin (\lambda K[1]+\mu K[1]))}{2 (a \cos (\lambda K[1])+b \sin (\lambda K[1]))}dK[1]} (\cos (\lambda K[2]-\mu K[2])-\cos (\lambda K[2]+\mu K[2]))}{c \mu (d+K[3]) (b \cos (\lambda K[2]-\mu K[2])-b \cos (\lambda K[2]+\mu K[2])-a \sin (\lambda K[2]-\mu K[2])+a \sin (\lambda K[2]+\mu K[2]))}-\frac {e^{-\int _1^{K[2]}-\frac {\csc (\mu K[1]) (-2 d \cos (\lambda K[1]-\mu K[1])+2 d \cos (\lambda K[1]+\mu K[1])+c \sin (\lambda K[1]-\mu K[1])+c \sin (\lambda K[1]+\mu K[1]))}{2 (a \cos (\lambda K[1])+b \sin (\lambda K[1]))}dK[1]} (-d \cos (\lambda K[2]-\mu K[2])+K[3] \cos (\lambda K[2]-\mu K[2])+d \cos (\lambda K[2]+\mu K[2])-\cos (\lambda K[2]+\mu K[2]) K[3]+c \sin (\lambda K[2]-\mu K[2])+c \sin (\lambda K[2]+\mu K[2]))}{c \mu (d+K[3])^2 (b \cos (\lambda K[2]-\mu K[2])-b \cos (\lambda K[2]+\mu K[2])-a \sin (\lambda K[2]-\mu K[2])+a \sin (\lambda K[2]+\mu K[2]))}\right )dK[2]-\frac {e^{-\int _1^x-\frac {\csc (\mu K[1]) (-2 d \cos (\lambda K[1]-\mu K[1])+2 d \cos (\lambda K[1]+\mu K[1])+c \sin (\lambda K[1]-\mu K[1])+c \sin (\lambda K[1]+\mu K[1]))}{2 (a \cos (\lambda K[1])+b \sin (\lambda K[1]))}dK[1]}}{c \mu (d+K[3])^2}\right )dK[3]=c_1,y(x)\right ] \]