Internal problem ID [9756]
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: 1.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {y^{\prime }-\alpha y^{2}-\beta -\gamma \sin \left (\lambda x \right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 110
dsolve(diff(y(x),x)=alpha*y(x)^2+beta+gamma*sin(lambda*x),y(x), singsol=all)
\[ y \relax (x ) = -\frac {\lambda \left (c_{1} \MathieuSPrime \left (\frac {4 \alpha \beta }{\lambda ^{2}}, -\frac {2 \alpha \gamma }{\lambda ^{2}}, -\frac {\pi }{4}+\frac {\lambda x}{2}\right )+\MathieuCPrime \left (\frac {4 \alpha \beta }{\lambda ^{2}}, -\frac {2 \alpha \gamma }{\lambda ^{2}}, -\frac {\pi }{4}+\frac {\lambda x}{2}\right )\right )}{2 \alpha \left (c_{1} \MathieuS \left (\frac {4 \alpha \beta }{\lambda ^{2}}, -\frac {2 \alpha \gamma }{\lambda ^{2}}, -\frac {\pi }{4}+\frac {\lambda x}{2}\right )+\MathieuC \left (\frac {4 \alpha \beta }{\lambda ^{2}}, -\frac {2 \alpha \gamma }{\lambda ^{2}}, -\frac {\pi }{4}+\frac {\lambda x}{2}\right )\right )} \]
✓ Solution by Mathematica
Time used: 0.327 (sec). Leaf size: 191
DSolve[y'[x]==\[Alpha]*y[x]^2+\[Beta]+\[Gamma]*Sin[\[Lambda]*x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\lambda \left (\text {MathieuSPrime}\left [\frac {4 \alpha \beta }{\lambda ^2},-\frac {2 \alpha \gamma }{\lambda ^2},\frac {1}{4} (\pi -2 \lambda x)\right ]+c_1 \text {MathieuCPrime}\left [\frac {4 \alpha \beta }{\lambda ^2},-\frac {2 \alpha \gamma }{\lambda ^2},\frac {1}{4} (2 \lambda x-\pi )\right ]\right )}{2 \alpha \left (\text {MathieuS}\left [\frac {4 \alpha \beta }{\lambda ^2},-\frac {2 \alpha \gamma }{\lambda ^2},\frac {1}{4} (2 \lambda x-\pi )\right ]+c_1 \text {MathieuC}\left [\frac {4 \alpha \beta }{\lambda ^2},-\frac {2 \alpha \gamma }{\lambda ^2},\frac {1}{4} (\pi -2 \lambda x)\right ]\right )} \\ y(x)\to \frac {\lambda \text {MathieuCPrime}\left [\frac {4 \alpha \beta }{\lambda ^2},-\frac {2 \alpha \gamma }{\lambda ^2},\frac {1}{4} (\pi -2 \lambda x)\right ]}{2 \alpha \text {MathieuC}\left [\frac {4 \alpha \beta }{\lambda ^2},-\frac {2 \alpha \gamma }{\lambda ^2},\frac {1}{4} (\pi -2 \lambda x)\right ]} \\ \end{align*}