Internal problem ID [9772]
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: 17.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {y^{\prime }-y^{2}-a \cos \left (\beta x \right ) y-a b \cos \left (\beta x \right )+b^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 48
dsolve(diff(y(x),x)=y(x)^2+a*cos(beta*x)*y(x)+a*b*cos(beta*x)-b^2,y(x), singsol=all)
\[ y \relax (x ) = -b -\frac {{\mathrm e}^{\frac {a \sin \left (\beta x \right )}{\beta }-2 b x}}{\int {\mathrm e}^{\frac {a \sin \left (\beta x \right )}{\beta }-2 b x}d x -c_{1}} \]
✓ Solution by Mathematica
Time used: 4.443 (sec). Leaf size: 183
DSolve[y'[x]==y[x]^2+a*Cos[\[Beta]*x]*y[x]+a*b*Cos[\[Beta]*x]-b^2,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^x\frac {e^{\frac {a \sin (\beta K[1])}{\beta }-2 b K[1]} (-b+a \cos (\beta K[1])+y(x))}{a \beta (b+y(x))}dK[1]+\int _1^{y(x)}\left (-\int _1^x\left (\frac {e^{\frac {a \sin (\beta K[1])}{\beta }-2 b K[1]}}{a \beta (b+K[2])}-\frac {e^{\frac {a \sin (\beta K[1])}{\beta }-2 b K[1]} (-b+a \cos (\beta K[1])+K[2])}{a \beta (b+K[2])^2}\right )dK[1]-\frac {e^{\frac {a \sin (x \beta )}{\beta }-2 b x}}{a \beta (b+K[2])^2}\right )dK[2]=c_1,y(x)\right ] \]