Internal problem ID [10477]
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: 19.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }-y^{2}=3 \lambda a -\lambda ^{2}-a \left (a +\lambda \right ) \tanh \left (\lambda x \right )^{2}} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 245
dsolve(diff(y(x),x)=y(x)^2+3*a*lambda-lambda^2-a*(a+lambda)*tanh(lambda*x)^2,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left (\left (-c_{1} a -c_{1} \lambda \right ) \operatorname {LegendreQ}\left (\frac {a}{\lambda }, \frac {a -\lambda }{\lambda }, \tanh \left (\lambda x \right )\right )+\left (-a -\lambda \right ) \operatorname {LegendreP}\left (\frac {a}{\lambda }, \frac {a -\lambda }{\lambda }, \tanh \left (\lambda x \right )\right )\right ) \tanh \left (\lambda x \right )}{c_{1} \operatorname {LegendreQ}\left (\frac {a}{\lambda }, \frac {a -\lambda }{\lambda }, \tanh \left (\lambda x \right )\right )+\operatorname {LegendreP}\left (\frac {a}{\lambda }, \frac {a -\lambda }{\lambda }, \tanh \left (\lambda x \right )\right )}+\frac {2 c_{1} \lambda \operatorname {LegendreQ}\left (\frac {a +\lambda }{\lambda }, \frac {a -\lambda }{\lambda }, \tanh \left (\lambda x \right )\right )}{c_{1} \operatorname {LegendreQ}\left (\frac {a}{\lambda }, \frac {a -\lambda }{\lambda }, \tanh \left (\lambda x \right )\right )+\operatorname {LegendreP}\left (\frac {a}{\lambda }, \frac {a -\lambda }{\lambda }, \tanh \left (\lambda x \right )\right )}+\frac {2 \operatorname {LegendreP}\left (\frac {a +\lambda }{\lambda }, \frac {a -\lambda }{\lambda }, \tanh \left (\lambda x \right )\right ) \lambda }{c_{1} \operatorname {LegendreQ}\left (\frac {a}{\lambda }, \frac {a -\lambda }{\lambda }, \tanh \left (\lambda x \right )\right )+\operatorname {LegendreP}\left (\frac {a}{\lambda }, \frac {a -\lambda }{\lambda }, \tanh \left (\lambda x \right )\right )} \]
✓ Solution by Mathematica
Time used: 12.804 (sec). Leaf size: 631
DSolve[y'[x]==y[x]^2+3*a*\[Lambda]-\[Lambda]^2-a*(a+\[Lambda])*Tanh[\[Lambda]*x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {-\lambda (a-2 \lambda ) \left (e^{2 \lambda x}-1\right ) \left (e^{2 \lambda x}+1\right )^{\frac {2 a}{\lambda }} \left (\frac {1}{e^{2 \lambda x}-1}+1\right )^{a/\lambda } \left (a \left (4 e^{2 \lambda x}+e^{4 \lambda x}-1\right )+\lambda -\lambda e^{4 \lambda x}\right ) \operatorname {AppellF1}\left (1-\frac {a}{\lambda },\frac {a}{\lambda },-\frac {2 a}{\lambda },2-\frac {a}{\lambda },\frac {1}{1-e^{2 x \lambda }},-\frac {2}{-1+e^{2 x \lambda }}\right )+(a-\lambda ) \left (8 a \lambda e^{2 \lambda x} \left (\frac {1}{e^{2 \lambda x}-1}+1\right )^{a/\lambda } \left (e^{2 \lambda x}+1\right )^{\frac {2 a}{\lambda }+1} \operatorname {AppellF1}\left (2-\frac {a}{\lambda },\frac {a}{\lambda },1-\frac {2 a}{\lambda },3-\frac {a}{\lambda },\frac {1}{1-e^{2 x \lambda }},-\frac {2}{-1+e^{2 x \lambda }}\right )-2 a \lambda e^{2 \lambda x} \left (\frac {1}{e^{2 \lambda x}-1}+1\right )^{a/\lambda } \left (e^{2 \lambda x}+1\right )^{\frac {2 a}{\lambda }+1} \operatorname {AppellF1}\left (2-\frac {a}{\lambda },\frac {a+\lambda }{\lambda },-\frac {2 a}{\lambda },3-\frac {a}{\lambda },\frac {1}{1-e^{2 x \lambda }},-\frac {2}{-1+e^{2 x \lambda }}\right )+c_1 (2 \lambda -a) \left (e^{2 \lambda x}-1\right )^2 \left (e^{2 \lambda x}\right )^{a/\lambda } \left (\frac {2}{e^{2 \lambda x}-1}+1\right )^{\frac {2 a}{\lambda }} \left (\lambda \left (e^{2 \lambda x}+1\right )^2-a \left (e^{2 \lambda x}-1\right )^2\right )\right )}{(2 \lambda -a) \left (e^{2 \lambda x}-1\right )^2 \left (e^{2 \lambda x}+1\right ) \left (-\lambda \left (e^{2 \lambda x}+1\right )^{\frac {2 a}{\lambda }} \left (\frac {1}{e^{2 \lambda x}-1}+1\right )^{a/\lambda } \operatorname {AppellF1}\left (1-\frac {a}{\lambda },\frac {a}{\lambda },-\frac {2 a}{\lambda },2-\frac {a}{\lambda },\frac {1}{1-e^{2 x \lambda }},-\frac {2}{-1+e^{2 x \lambda }}\right )+c_1 (\lambda -a) \left (e^{2 \lambda x}-1\right ) \left (e^{2 \lambda x}\right )^{a/\lambda } \left (\frac {2}{e^{2 \lambda x}-1}+1\right )^{\frac {2 a}{\lambda }}\right )} y(x)\to \frac {a \left (e^{2 \lambda x}-1\right )^2-\lambda \left (e^{2 \lambda x}+1\right )^2}{e^{4 \lambda x}-1} \end{align*}