Internal problem ID [9821]
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: 7.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_symmetry_[F(x),G(x)]], _Riccati]
Solve \begin {gather*} \boxed {y^{\prime }-\lambda \arcsin \relax (x )^{n} \left (y-a \,x^{m}-b \right )^{2}-a m \,x^{m -1}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 50
dsolve(diff(y(x),x)=lambda*arcsin(x)^n*(y(x)-a*x^m-b)^2+a*m*x^(m-1),y(x), singsol=all)
\[ y \relax (x ) = -\frac {\left (-2 \arcsin \relax (x )^{n} x^{m} a \lambda -2 \arcsin \relax (x )^{n} \lambda b \right ) \arcsin \relax (x )^{-n}}{2 \lambda }+\frac {1}{c_{1}-\left (\int \arcsin \relax (x )^{n} \lambda d x \right )} \]
✓ Solution by Mathematica
Time used: 3.481 (sec). Leaf size: 87
DSolve[y'[x]==\[Lambda]*ArcSin[x]^n*(y[x]-a*x^m-b)^2+a*m*x^(m-1),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to a x^m+\frac {1}{\frac {1}{2} i \lambda \text {ArcSin}(x)^n \left (\text {ArcSin}(x)^2\right )^{-n} \left ((i \text {ArcSin}(x))^n \Gamma (n+1,-i \text {ArcSin}(x))-(-i \text {ArcSin}(x))^n \Gamma (n+1,i \text {ArcSin}(x))\right )+c_1}+b \\ y(x)\to a x^m+b \\ \end{align*}