Internal problem ID [10479]
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]
\[ \boxed {\left (\tanh \left (\lambda x \right ) a +b \right ) y^{\prime }-y^{2}-c \tanh \left (\mu x \right ) y=-d^{2}+c d \tanh \left (\mu x \right )} \]
✗ Solution by Maple
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)
\[ \text {No solution found} \]
✓ 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 ] \]