61.10.4 problem 17

Internal problem ID [12191]
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
Date solved : Tuesday, January 28, 2025 at 01:01:12 AM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&=y^{2}+a \cos \left (\beta x \right ) y+a b \cos \left (\beta x \right )-b^{2} \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 73

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)
 

\[ y = \frac {b \left (\int {\mathrm e}^{\frac {-2 b \beta x +a \sin \left (\beta x \right )}{\beta }}d x \right )-c_{1} b +{\mathrm e}^{\frac {-2 b \beta x +a \sin \left (\beta x \right )}{\beta }}}{-\int {\mathrm e}^{\frac {-2 b \beta x +a \sin \left (\beta x \right )}{\beta }}d x +c_{1}} \]

✓ Solution by Mathematica

Time used: 3.292 (sec). Leaf size: 208

DSolve[D[y[x],x]==y[x]^2+a*Cos[\[Beta]*x]*y[x]+a*b*Cos[\[Beta]*x]-b^2,y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\int _1^x\frac {\exp \left (-\int _1^{K[2]}(2 b-a \cos (\beta K[1]))dK[1]\right ) (-b+a \cos (\beta K[2])+y(x))}{a \beta (b+y(x))}dK[2]+\int _1^{y(x)}\left (-\int _1^x\left (\frac {\exp \left (-\int _1^{K[2]}(2 b-a \cos (\beta K[1]))dK[1]\right )}{a \beta (b+K[3])}-\frac {\exp \left (-\int _1^{K[2]}(2 b-a \cos (\beta K[1]))dK[1]\right ) (-b+a \cos (\beta K[2])+K[3])}{a \beta (b+K[3])^2}\right )dK[2]-\frac {\exp \left (-\int _1^x(2 b-a \cos (\beta K[1]))dK[1]\right )}{a \beta (b+K[3])^2}\right )dK[3]=c_1,y(x)\right ] \]