Internal problem ID [10560]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 1, section 1.2. Riccati Equation. subsection 1.2.7-1. Equations containing
arcsine.
Problem number: 3.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }+\left (k +1\right ) x^{k} y^{2}-\lambda \arcsin \left (x \right )^{n} \left (x^{k +1} y-1\right )=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 180
dsolve(diff(y(x),x)=-(k+1)*x^k*y(x)^2+lambda*arcsin(x)^n*(x^(k+1)*y(x)-1),y(x), singsol=all)
\[ y \left (x \right ) = \frac {x^{-1-k} \left (x^{1+k} {\mathrm e}^{\int \frac {\arcsin \left (x \right )^{n} x^{1+k} \lambda x -2 k -2}{x}d x}+\left (\int x^{k} {\mathrm e}^{\lambda \left (\int \arcsin \left (x \right )^{n} x^{1+k}d x \right )-2 \left (\int \frac {1}{x}d x \right ) \left (1+k \right )}d x \right ) k +\int x^{k} {\mathrm e}^{\lambda \left (\int \arcsin \left (x \right )^{n} x^{1+k}d x \right )-2 \left (\int \frac {1}{x}d x \right ) \left (1+k \right )}d x +c_{1} \right )}{\left (\int x^{k} {\mathrm e}^{\lambda \left (\int \arcsin \left (x \right )^{n} x^{1+k}d x \right )-2 \left (\int \frac {1}{x}d x \right ) \left (1+k \right )}d x \right ) k +\int x^{k} {\mathrm e}^{\lambda \left (\int \arcsin \left (x \right )^{n} x^{1+k}d x \right )-2 \left (\int \frac {1}{x}d x \right ) \left (1+k \right )}d x +c_{1}} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y'[x]==-(k+1)*x^k*y[x]^2+\[Lambda]*ArcSin[x]^n*(x^(k+1)*y[x]-1),y[x],x,IncludeSingularSolutions -> True]
Not solved