Internal problem ID [10510]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 1, section 1.2. Riccati Equation. subsection 1.2.6-1. Equations with
sine
Problem number: 12.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {\left (a \sin \left (\lambda x \right )+b \right ) y^{\prime }-y^{2}-c \sin \left (x \mu \right ) y=-d^{2}+c d \sin \left (x \mu \right )} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 265
dsolve((a*sin(lambda*x)+b)*diff(y(x),x)=y(x)^2+c*sin(mu*x)*y(x)-d^2+c*d*sin(mu*x),y(x), singsol=all)
\[ y \left (x \right ) = \frac {-d \left (\int \frac {{\mathrm e}^{\frac {c \left (\int \frac {\sin \left (x \mu \right )}{a \sin \left (x \lambda \right )+b}d x \right ) \sqrt {-a^{2}+b^{2}}\, \lambda -4 d \arctan \left (\frac {\tan \left (\frac {x \lambda }{2}\right ) b +a}{\sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, \lambda }}}{a \sin \left (x \lambda \right )+b}d x \right )+d c_{1} -{\mathrm e}^{\frac {c \left (\int \frac {\sin \left (x \mu \right )}{a \sin \left (x \lambda \right )+b}d x \right ) \sqrt {-a^{2}+b^{2}}\, \lambda -4 d \arctan \left (\frac {\tan \left (\frac {x \lambda }{2}\right ) b +a}{\sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, \lambda }}}{\int \frac {{\mathrm e}^{\frac {c \left (\int \frac {\sin \left (x \mu \right )}{a \sin \left (x \lambda \right )+b}d x \right ) \sqrt {-a^{2}+b^{2}}\, \lambda -4 d \arctan \left (\frac {\tan \left (\frac {x \lambda }{2}\right ) b +a}{\sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, \lambda }}}{a \sin \left (x \lambda \right )+b}d x -c_{1}} \]
✓ Solution by Mathematica
Time used: 15.846 (sec). Leaf size: 289
DSolve[(a*Sin[\[Lambda]*x]+b)*y'[x]==y[x]^2+c*Sin[\[Mu]*x]*y[x]-d^2+c*d*Sin[\[Mu]*x],y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^x-\frac {\exp \left (-\int _1^{K[2]}\frac {2 d-c \sin (\mu K[1])}{b+a \sin (\lambda K[1])}dK[1]\right ) (-d+c \sin (\mu K[2])+y(x))}{c \mu (b+a \sin (\lambda K[2])) (d+y(x))}dK[2]+\int _1^{y(x)}\left (\frac {\exp \left (-\int _1^x\frac {2 d-c \sin (\mu K[1])}{b+a \sin (\lambda K[1])}dK[1]\right )}{c \mu (d+K[3])^2}-\int _1^x\left (\frac {\exp \left (-\int _1^{K[2]}\frac {2 d-c \sin (\mu K[1])}{b+a \sin (\lambda K[1])}dK[1]\right ) (-d+K[3]+c \sin (\mu K[2]))}{c \mu (d+K[3])^2 (b+a \sin (\lambda K[2]))}-\frac {\exp \left (-\int _1^{K[2]}\frac {2 d-c \sin (\mu K[1])}{b+a \sin (\lambda K[1])}dK[1]\right )}{c \mu (d+K[3]) (b+a \sin (\lambda K[2]))}\right )dK[2]\right )dK[3]=c_1,y(x)\right ] \]