Internal problem ID [9726]
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: 21.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {\left (\tanh \left (\lambda x \right ) a +b \right ) y^{\prime }-y^{2}-c \tanh \left (\mu x \right ) y+d^{2}-c d \tanh \left (\mu x \right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 217
dsolve((a*tanh(lambda*x)+b)*diff(y(x),x)=y(x)^2+c*tanh(mu*x)*y(x)-d^2+c*d*tanh(mu*x),y(x), singsol=all)
\[ y \relax (x ) = -d -\frac {{\mathrm e}^{\int \frac {c \tanh \left (\mu x \right )}{a \tanh \left (\lambda x \right )+b}d x} \left (a \tanh \left (\lambda x \right )+b \right )^{-\frac {2 a d}{\lambda \left (a +b \right ) \left (a -b \right )}} \left (\tanh \left (\lambda x \right )-1\right )^{\frac {d}{\lambda \left (a +b \right )}} \left (\tanh \left (\lambda x \right )+1\right )^{\frac {d}{\left (a -b \right ) \lambda }}}{\int \frac {{\mathrm e}^{\int \frac {c \tanh \left (\mu x \right )}{a \tanh \left (\lambda x \right )+b}d x} \left (a \tanh \left (\lambda x \right )+b \right )^{-\frac {2 a d}{\lambda \left (a +b \right ) \left (a -b \right )}} \left (\tanh \left (\lambda x \right )-1\right )^{\frac {d}{\lambda \left (a +b \right )}} \left (\tanh \left (\lambda x \right )+1\right )^{\frac {d}{\left (a -b \right ) \lambda }}}{a \tanh \left (\lambda x \right )+b}d x -c_{1}} \]
✓ Solution by Mathematica
Time used: 75.864 (sec). Leaf size: 800
DSolve[(a*Tanh[\[Lambda]*x]+b)*y'[x]==y[x]^2+c*Tanh[\[Mu]*x]*y[x]-d^2+c*d*Tanh[\[Mu]*x],y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^x\frac {e^{-\int _1^{K[6]}\frac {\text {sech}(\mu K[5]) (2 d \cosh (\lambda K[5]-\mu K[5])+2 d \cosh (\lambda K[5]+\mu K[5])+c \sinh (\lambda K[5]-\mu K[5])-c \sinh (\lambda K[5]+\mu K[5]))}{2 (b \cosh (\lambda K[5])+a \sinh (\lambda K[5]))}dK[5]} (d \cosh (\lambda K[6]-\mu K[6])-y(x) \cosh (\lambda K[6]-\mu K[6])+d \cosh (\lambda K[6]+\mu K[6])+c \sinh (\lambda K[6]-\mu K[6])-c \sinh (\lambda K[6]+\mu K[6])-\cosh (\lambda K[6]+\mu K[6]) y(x))}{c \mu (b \cosh (\lambda K[6]-\mu K[6])+b \cosh (\lambda K[6]+\mu K[6])+a \sinh (\lambda K[6]-\mu K[6])+a \sinh (\lambda K[6]+\mu K[6])) (d+y(x))}dK[6]+\int _1^{y(x)}\left (\frac {e^{-\int _1^x\frac {\text {sech}(\mu K[5]) (2 d \cosh (\lambda K[5]-\mu K[5])+2 d \cosh (\lambda K[5]+\mu K[5])+c \sinh (\lambda K[5]-\mu K[5])-c \sinh (\lambda K[5]+\mu K[5]))}{2 (b \cosh (\lambda K[5])+a \sinh (\lambda K[5]))}dK[5]}}{c \mu (d+K[7])^2}-\int _1^x\left (\frac {e^{-\int _1^{K[6]}\frac {\text {sech}(\mu K[5]) (2 d \cosh (\lambda K[5]-\mu K[5])+2 d \cosh (\lambda K[5]+\mu K[5])+c \sinh (\lambda K[5]-\mu K[5])-c \sinh (\lambda K[5]+\mu K[5]))}{2 (b \cosh (\lambda K[5])+a \sinh (\lambda K[5]))}dK[5]} (-\cosh (\lambda K[6]-\mu K[6])-\cosh (\lambda K[6]+\mu K[6]))}{c \mu (d+K[7]) (b \cosh (\lambda K[6]-\mu K[6])+b \cosh (\lambda K[6]+\mu K[6])+a \sinh (\lambda K[6]-\mu K[6])+a \sinh (\lambda K[6]+\mu K[6]))}-\frac {e^{-\int _1^{K[6]}\frac {\text {sech}(\mu K[5]) (2 d \cosh (\lambda K[5]-\mu K[5])+2 d \cosh (\lambda K[5]+\mu K[5])+c \sinh (\lambda K[5]-\mu K[5])-c \sinh (\lambda K[5]+\mu K[5]))}{2 (b \cosh (\lambda K[5])+a \sinh (\lambda K[5]))}dK[5]} (d \cosh (\lambda K[6]-\mu K[6])-K[7] \cosh (\lambda K[6]-\mu K[6])+d \cosh (\lambda K[6]+\mu K[6])-\cosh (\lambda K[6]+\mu K[6]) K[7]+c \sinh (\lambda K[6]-\mu K[6])-c \sinh (\lambda K[6]+\mu K[6]))}{c \mu (d+K[7])^2 (b \cosh (\lambda K[6]-\mu K[6])+b \cosh (\lambda K[6]+\mu K[6])+a \sinh (\lambda K[6]-\mu K[6])+a \sinh (\lambda K[6]+\mu K[6]))}\right )dK[6]\right )dK[7]=c_1,y(x)\right ] \]