Internal problem ID [9711]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second
edition
Section: Chapter 1, section 1.2. Riccati Equation. subsection 1.2.4-1. Equations with hyperbolic sine
and cosine
Problem number: 6.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {\left (\sinh \left (\lambda x \right ) a +b \right ) y^{\prime }-y^{2}-c \sinh \left (\mu x \right ) y+d^{2}-c d \sinh \left (\mu x \right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 147
dsolve((a*sinh(lambda*x)+b)*diff(y(x),x)=y(x)^2+c*sinh(mu*x)*y(x)-d^2+c*d*sinh(mu*x),y(x), singsol=all)
\[ y \relax (x ) = -d -\frac {{\mathrm e}^{\int \frac {\sinh \left (\mu x \right ) c}{\sinh \left (\lambda x \right ) a +b}d x -\frac {4 d \arctanh \left (\frac {2 b \tanh \left (\frac {\lambda x}{2}\right )-2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{\lambda \sqrt {a^{2}+b^{2}}}}}{\int \frac {{\mathrm e}^{\int \frac {\sinh \left (\mu x \right ) c}{\sinh \left (\lambda x \right ) a +b}d x -\frac {4 d \arctanh \left (\frac {2 b \tanh \left (\frac {\lambda x}{2}\right )-2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{\lambda \sqrt {a^{2}+b^{2}}}}}{\sinh \left (\lambda x \right ) a +b}d x -c_{1}} \]
✓ Solution by Mathematica
Time used: 14.184 (sec). Leaf size: 289
DSolve[(a*Sinh[\[Lambda]*x]+b)*y'[x]==y[x]^2+c*Sinh[\[Mu]*x]*y[x]-d^2+c*d*Sinh[\[Mu]*x],y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^x-\frac {\exp \left (-\int _1^{K[6]}\frac {2 d-c \sinh (\mu K[5])}{b+a \sinh (\lambda K[5])}dK[5]\right ) (-d+c \sinh (\mu K[6])+y(x))}{c \mu (b+a \sinh (\lambda K[6])) (d+y(x))}dK[6]+\int _1^{y(x)}\left (\frac {\exp \left (-\int _1^x\frac {2 d-c \sinh (\mu K[5])}{b+a \sinh (\lambda K[5])}dK[5]\right )}{c \mu (d+K[7])^2}-\int _1^x\left (\frac {\exp \left (-\int _1^{K[6]}\frac {2 d-c \sinh (\mu K[5])}{b+a \sinh (\lambda K[5])}dK[5]\right ) (-d+K[7]+c \sinh (\mu K[6]))}{c \mu (d+K[7])^2 (b+a \sinh (\lambda K[6]))}-\frac {\exp \left (-\int _1^{K[6]}\frac {2 d-c \sinh (\mu K[5])}{b+a \sinh (\lambda K[5])}dK[5]\right )}{c \mu (d+K[7]) (b+a \sinh (\lambda K[6]))}\right )dK[6]\right )dK[7]=c_1,y(x)\right ] \]