Internal problem ID [9767]
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]
Solve \begin {gather*} \boxed {\left (\sin \left (\lambda x \right ) a +b \right ) y^{\prime }-y^{2}-c \sin \left (\mu x \right ) y+d^{2}-c d \sin \left (\mu x \right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.013 (sec). Leaf size: 155
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)
\[ y \relax (x ) = -d -\frac {{\mathrm e}^{\int \frac {c \sin \left (\mu x \right )}{a \sin \left (\lambda x \right )+b}d x -\frac {4 d \arctan \left (\frac {2 \tan \left (\frac {\lambda x}{2}\right ) b +2 a}{2 \sqrt {-a^{2}+b^{2}}}\right )}{\lambda \sqrt {-a^{2}+b^{2}}}}}{\int \frac {{\mathrm e}^{\int \frac {c \sin \left (\mu x \right )}{a \sin \left (\lambda x \right )+b}d x -\frac {4 d \arctan \left (\frac {2 \tan \left (\frac {\lambda x}{2}\right ) b +2 a}{2 \sqrt {-a^{2}+b^{2}}}\right )}{\lambda \sqrt {-a^{2}+b^{2}}}}}{a \sin \left (\lambda x \right )+b}d x -c_{1}} \]
✓ Solution by Mathematica
Time used: 8.28 (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]
\[ \text {Solve}\left [\int _1^x-\frac {\exp \left (-\int _1^{K[6]}\frac {2 d-c \sin (\mu K[5])}{b+a \sin (\lambda K[5])}dK[5]\right ) (-d+c \sin (\mu K[6])+y(x))}{c \mu (b+a \sin (\lambda K[6])) (d+y(x))}dK[6]+\int _1^{y(x)}\left (\frac {\exp \left (-\int _1^x\frac {2 d-c \sin (\mu K[5])}{b+a \sin (\lambda K[5])}dK[5]\right )}{c \mu (d+K[7])^2}-\int _1^x\left (\frac {\exp \left (-\int _1^{K[6]}\frac {2 d-c \sin (\mu K[5])}{b+a \sin (\lambda K[5])}dK[5]\right ) (-d+K[7]+c \sin (\mu K[6]))}{c \mu (d+K[7])^2 (b+a \sin (\lambda K[6]))}-\frac {\exp \left (-\int _1^{K[6]}\frac {2 d-c \sin (\mu K[5])}{b+a \sin (\lambda K[5])}dK[5]\right )}{c \mu (d+K[7]) (b+a \sin (\lambda K[6]))}\right )dK[6]\right )dK[7]=c_1,y(x)\right ] \]