Internal problem ID [10483]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 1, section 1.2. Riccati Equation. subsection 1.2.4-2. Equations with hyperbolic
tangent and cotangent.
Problem number: 25.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {\left (a \coth \left (x \lambda \right )+b \right ) y^{\prime }-y^{2}-c \coth \left (\mu x \right ) y=-d^{2}+c d \coth \left (\mu x \right )} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 217
dsolve((a*coth(lambda*x)+b)*diff(y(x),x)=y(x)^2+c*coth(mu*x)*y(x)-d^2+c*d*coth(mu*x),y(x), singsol=all)
\[ y \left (x \right ) = -d -\frac {{\mathrm e}^{\int \frac {c \coth \left (\mu x \right )}{a \coth \left (\lambda x \right )+b}d x} \left (a \coth \left (\lambda x \right )+b \right )^{-\frac {2 a d}{\lambda \left (a -b \right ) \left (a +b \right )}} \left (\coth \left (\lambda x \right )-1\right )^{\frac {d}{\lambda \left (a +b \right )}} \left (\coth \left (\lambda x \right )+1\right )^{\frac {d}{\lambda \left (a -b \right )}}}{\int \frac {{\mathrm e}^{\int \frac {c \coth \left (\mu x \right )}{a \coth \left (\lambda x \right )+b}d x} \left (a \coth \left (\lambda x \right )+b \right )^{-\frac {2 a d}{\lambda \left (a -b \right ) \left (a +b \right )}} \left (\coth \left (\lambda x \right )-1\right )^{\frac {d}{\lambda \left (a +b \right )}} \left (\coth \left (\lambda x \right )+1\right )^{\frac {d}{\lambda \left (a -b \right )}}}{a \coth \left (\lambda x \right )+b}d x -c_{1}} \]
✓ Solution by Mathematica
Time used: 153.106 (sec). Leaf size: 808
DSolve[(a*Coth[\[Lambda]*x]+b)*y'[x]==y[x]^2+c*Coth[\[Mu]*x]*y[x]-d^2+c*d*Coth[\[Mu]*x],y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^x-\frac {e^{-\int _1^{K[2]}\frac {\text {csch}(\mu K[1]) (-2 d \cosh (\lambda K[1]-\mu K[1])+2 d \cosh (\lambda K[1]+\mu K[1])-c \sinh (\lambda K[1]-\mu K[1])-c \sinh (\lambda K[1]+\mu K[1]))}{2 (a \cosh (\lambda K[1])+b \sinh (\lambda K[1]))}dK[1]} (d \cosh (\lambda K[2]-\mu K[2])-y(x) \cosh (\lambda K[2]-\mu K[2])-d \cosh (\lambda K[2]+\mu K[2])+c \sinh (\lambda K[2]-\mu K[2])+c \sinh (\lambda K[2]+\mu K[2])+\cosh (\lambda K[2]+\mu K[2]) y(x))}{c \mu (b \cosh (\lambda K[2]-\mu K[2])-b \cosh (\lambda K[2]+\mu K[2])+a \sinh (\lambda K[2]-\mu K[2])-a \sinh (\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 {\text {csch}(\mu K[1]) (-2 d \cosh (\lambda K[1]-\mu K[1])+2 d \cosh (\lambda K[1]+\mu K[1])-c \sinh (\lambda K[1]-\mu K[1])-c \sinh (\lambda K[1]+\mu K[1]))}{2 (a \cosh (\lambda K[1])+b \sinh (\lambda K[1]))}dK[1]} (d \cosh (\lambda K[2]-\mu K[2])-K[3] \cosh (\lambda K[2]-\mu K[2])-d \cosh (\lambda K[2]+\mu K[2])+\cosh (\lambda K[2]+\mu K[2]) K[3]+c \sinh (\lambda K[2]-\mu K[2])+c \sinh (\lambda K[2]+\mu K[2]))}{c \mu (d+K[3])^2 (b \cosh (\lambda K[2]-\mu K[2])-b \cosh (\lambda K[2]+\mu K[2])+a \sinh (\lambda K[2]-\mu K[2])-a \sinh (\lambda K[2]+\mu K[2]))}-\frac {e^{-\int _1^{K[2]}\frac {\text {csch}(\mu K[1]) (-2 d \cosh (\lambda K[1]-\mu K[1])+2 d \cosh (\lambda K[1]+\mu K[1])-c \sinh (\lambda K[1]-\mu K[1])-c \sinh (\lambda K[1]+\mu K[1]))}{2 (a \cosh (\lambda K[1])+b \sinh (\lambda K[1]))}dK[1]} (\cosh (\lambda K[2]+\mu K[2])-\cosh (\lambda K[2]-\mu K[2]))}{c \mu (d+K[3]) (b \cosh (\lambda K[2]-\mu K[2])-b \cosh (\lambda K[2]+\mu K[2])+a \sinh (\lambda K[2]-\mu K[2])-a \sinh (\lambda K[2]+\mu K[2]))}\right )dK[2]-\frac {e^{-\int _1^x\frac {\text {csch}(\mu K[1]) (-2 d \cosh (\lambda K[1]-\mu K[1])+2 d \cosh (\lambda K[1]+\mu K[1])-c \sinh (\lambda K[1]-\mu K[1])-c \sinh (\lambda K[1]+\mu K[1]))}{2 (a \cosh (\lambda K[1])+b \sinh (\lambda K[1]))}dK[1]}}{c \mu (d+K[3])^2}\right )dK[3]=c_1,y(x)\right ] \]