Internal problem ID [9765]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second
edition
Section: Chapter 1, section 1.2. Riccati Equation. subsection 1.2.6-2. Equations with cosine.
Problem number: 14.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }-\alpha y^{2}-\beta -\gamma \cos \left (\lambda x \right )=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 94
dsolve(diff(y(x),x)=alpha*y(x)^2+beta+gamma*cos(lambda*x),y(x), singsol=all)
\[ y \left (x \right ) = -\frac {\lambda \left (c_{1} \operatorname {MathieuSPrime}\left (\frac {4 \alpha \beta }{\lambda ^{2}}, -\frac {2 \gamma \alpha }{\lambda ^{2}}, \frac {\lambda x}{2}\right )+\operatorname {MathieuCPrime}\left (\frac {4 \alpha \beta }{\lambda ^{2}}, -\frac {2 \gamma \alpha }{\lambda ^{2}}, \frac {\lambda x}{2}\right )\right )}{2 \alpha \left (c_{1} \operatorname {MathieuS}\left (\frac {4 \alpha \beta }{\lambda ^{2}}, -\frac {2 \gamma \alpha }{\lambda ^{2}}, \frac {\lambda x}{2}\right )+\operatorname {MathieuC}\left (\frac {4 \alpha \beta }{\lambda ^{2}}, -\frac {2 \gamma \alpha }{\lambda ^{2}}, \frac {\lambda x}{2}\right )\right )} \]
✓ Solution by Mathematica
Time used: 0.314 (sec). Leaf size: 163
DSolve[y'[x]==\[Alpha]*y[x]^2+\[Beta]+\[Gamma]*Cos[\[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 {\lambda x}{2}\right ]+c_1 \text {MathieuCPrime}\left [\frac {4 \alpha \beta }{\lambda ^2},-\frac {2 \alpha \gamma }{\lambda ^2},\frac {\lambda x}{2}\right ]\right )}{2 \alpha \left (\text {MathieuS}\left [\frac {4 \alpha \beta }{\lambda ^2},-\frac {2 \alpha \gamma }{\lambda ^2},\frac {\lambda x}{2}\right ]+c_1 \text {MathieuC}\left [\frac {4 \alpha \beta }{\lambda ^2},-\frac {2 \alpha \gamma }{\lambda ^2},\frac {\lambda x}{2}\right ]\right )} \\ y(x)\to -\frac {\lambda \text {MathieuCPrime}\left [\frac {4 \alpha \beta }{\lambda ^2},-\frac {2 \alpha \gamma }{\lambda ^2},\frac {\lambda x}{2}\right ]}{2 \alpha \text {MathieuC}\left [\frac {4 \alpha \beta }{\lambda ^2},-\frac {2 \alpha \gamma }{\lambda ^2},\frac {\lambda x}{2}\right ]} \\ \end{align*}