Internal problem ID [10464]
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]
\[ \boxed {\left (a \sinh \left (x \lambda \right )+b \right ) y^{\prime }-y^{2}-c \sinh \left (\mu x \right ) y=-d^{2}+c d \sinh \left (\mu x \right )} \]
✓ Solution by Maple
Time used: 0.016 (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)
\[ y \left (x \right ) = -d -\frac {{\mathrm e}^{\int \frac {c \sinh \left (\mu x \right )}{\sinh \left (\lambda x \right ) a +b}d x -\frac {4 d \,\operatorname {arctanh}\left (\frac {2 b \tanh \left (\frac {\lambda x}{2}\right )-2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{\lambda \sqrt {a^{2}+b^{2}}}}}{\int \frac {{\mathrm e}^{\int \frac {c \sinh \left (\mu x \right )}{\sinh \left (\lambda x \right ) a +b}d x -\frac {4 d \,\operatorname {arctanh}\left (\frac {2 b \tanh \left (\frac {\lambda x}{2}\right )-2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{\lambda \sqrt {a^{2}+b^{2}}}}}{\sinh \left (\lambda x \right ) a +b}d x -c_{1}} \]
✓ Solution by Mathematica
Time used: 28.506 (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]
\[ \text {Solve}\left [\int _1^x-\frac {\exp \left (-\int _1^{K[2]}\frac {2 d-c \sinh (\mu K[1])}{b+a \sinh (\lambda K[1])}dK[1]\right ) (-d+c \sinh (\mu K[2])+y(x))}{c \mu (b+a \sinh (\lambda K[2])) (d+y(x))}dK[2]+\int _1^{y(x)}\left (\frac {\exp \left (-\int _1^x\frac {2 d-c \sinh (\mu K[1])}{b+a \sinh (\lambda K[1])}dK[1]\right )}{c \mu (d+K[3])^2}-\int _1^x\left (\frac {\exp \left (-\int _1^{K[2]}\frac {2 d-c \sinh (\mu K[1])}{b+a \sinh (\lambda K[1])}dK[1]\right ) (-d+K[3]+c \sinh (\mu K[2]))}{c \mu (d+K[3])^2 (b+a \sinh (\lambda K[2]))}-\frac {\exp \left (-\int _1^{K[2]}\frac {2 d-c \sinh (\mu K[1])}{b+a \sinh (\lambda K[1])}dK[1]\right )}{c \mu (d+K[3]) (b+a \sinh (\lambda K[2]))}\right )dK[2]\right )dK[3]=c_1,y(x)\right ] \]