Internal problem ID [10517]
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: 9.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }+\left (k +1\right ) x^{k} y^{2}-a \,x^{k +1} \sin \left (x \right )^{m} y=-a \sin \left (x \right )^{m}} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 172
dsolve(diff(y(x),x)=-(k+1)*x^k*y(x)^2+a*x^(k+1)*sin(x)^m*y(x)-a*sin(x)^m,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left ({\mathrm e}^{\int \frac {\sin \left (x \right )^{m} x^{k} a \,x^{2}-2 k -2}{x}d x} x \,x^{k}+\left (\int x^{k} {\mathrm e}^{\int \frac {a \,x^{k +2} \sin \left (x \right )^{m}-2 k -2}{x}d x}d x \right ) k +\int x^{k} {\mathrm e}^{\int \frac {a \,x^{k +2} \sin \left (x \right )^{m}-2 k -2}{x}d x}d x -c_{1} \right ) x^{-k}}{x \left (\left (\int x^{k} {\mathrm e}^{\int \frac {a \,x^{k +2} \sin \left (x \right )^{m}-2 k -2}{x}d x}d x \right ) k +\int x^{k} {\mathrm e}^{\int \frac {a \,x^{k +2} \sin \left (x \right )^{m}-2 k -2}{x}d x}d x -c_{1} \right )} \]
✓ Solution by Mathematica
Time used: 16.483 (sec). Leaf size: 248
DSolve[y'[x]==-(k+1)*x^k*y[x]^2+a*x^(k+1)*Sin[x]^m*y[x]-a*Sin[x]^m,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {x^{-k-1} \left (c_1 x \exp \left (\int _1^x-\frac {-a \sin ^m(K[1]) K[1]^{k+2}+k+2}{K[1]}dK[1]\right )+c_1 (k+1) \int _1^x\exp \left (\int _1^{K[2]}-\frac {-a \sin ^m(K[1]) K[1]^{k+2}+k+2}{K[1]}dK[1]\right )dK[2]+k+1\right )}{(k+1) \left (1+c_1 \int _1^x\exp \left (\int _1^{K[2]}-\frac {-a \sin ^m(K[1]) K[1]^{k+2}+k+2}{K[1]}dK[1]\right )dK[2]\right )} y(x)\to \frac {x^{-k} \left (\frac {\exp \left (\int _1^x-\frac {-a \sin ^m(K[1]) K[1]^{k+2}+k+2}{K[1]}dK[1]\right )}{\int _1^x\exp \left (\int _1^{K[2]}-\frac {-a \sin ^m(K[1]) K[1]^{k+2}+k+2}{K[1]}dK[1]\right )dK[2]}+\frac {k+1}{x}\right )}{k+1} \end{align*}