Internal problem ID [10469]
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]
\[ \boxed {\left (a \tanh \left (\lambda x \right )+b \right ) y^{\prime }-y^{2}-c \tanh \left (x \mu \right ) y=c d \tanh \left (x \mu \right )-d^{2}} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 302
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)
\[ y \left (x \right ) = \frac {-{\mathrm e}^{c \left (\int \frac {\tanh \left (x \mu \right )}{a \tanh \left (x \lambda \right )+b}d x \right )} \left (\tanh \left (x \lambda \right )+1\right )^{\frac {d}{\lambda \left (a -b \right )}} \left (\tanh \left (x \lambda \right )-1\right )^{\frac {d}{\lambda \left (a +b \right )}} \left (a \tanh \left (x \lambda \right )+b \right )^{-\frac {2 a d}{\lambda \left (a^{2}-b^{2}\right )}}-d \left (\int \left (a \tanh \left (x \lambda \right )+b \right )^{\frac {\left (-a^{2}+b^{2}\right ) \lambda -2 a d}{\lambda \left (a^{2}-b^{2}\right )}} \left (\tanh \left (x \lambda \right )-1\right )^{\frac {d}{\lambda \left (a +b \right )}} \left (\tanh \left (x \lambda \right )+1\right )^{\frac {d}{\lambda \left (a -b \right )}} {\mathrm e}^{c \left (\int \frac {\tanh \left (x \mu \right )}{a \tanh \left (x \lambda \right )+b}d x \right )}d x -c_{1} \right )}{\int \left (a \tanh \left (x \lambda \right )+b \right )^{\frac {\left (-a^{2}+b^{2}\right ) \lambda -2 a d}{\lambda \left (a^{2}-b^{2}\right )}} \left (\tanh \left (x \lambda \right )-1\right )^{\frac {d}{\lambda \left (a +b \right )}} \left (\tanh \left (x \lambda \right )+1\right )^{\frac {d}{\lambda \left (a -b \right )}} {\mathrm e}^{c \left (\int \frac {\tanh \left (x \mu \right )}{a \tanh \left (x \lambda \right )+b}d x \right )}d x -c_{1}} \]
✓ Solution by Mathematica
Time used: 163.692 (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[2]}\frac {\text {sech}(\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 (b \cosh (\lambda K[1])+a \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 (\frac {e^{-\int _1^x\frac {\text {sech}(\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 (b \cosh (\lambda K[1])+a \sinh (\lambda K[1]))}dK[1]}}{c \mu (d+K[3])^2}-\int _1^x\left (\frac {e^{-\int _1^{K[2]}\frac {\text {sech}(\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 (b \cosh (\lambda K[1])+a \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]))}-\frac {e^{-\int _1^{K[2]}\frac {\text {sech}(\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 (b \cosh (\lambda K[1])+a \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]))}\right )dK[2]\right )dK[3]=c_1,y(x)\right ] \]