Internal problem ID [10583]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 1, section 1.2. Riccati Equation. subsection 1.2.7-2. Equations containing
arccosine.
Problem number: 16.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]
\[ \boxed {y^{\prime }-\lambda \arccos \left (x \right )^{n} \left (y-a \,x^{m}-b \right )^{2}=a m \,x^{m -1}} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 162
dsolve(diff(y(x),x)=lambda*arccos(x)^n*(y(x)-a*x^m-b)^2+a*m*x^(m-1),y(x), singsol=all)
\[ y \left (x \right ) = -\frac {\left (-2 a \,x^{m} \arccos \left (x \right )^{n} \lambda -2 \arccos \left (x \right )^{n} \lambda b \right ) \arccos \left (x \right )^{-n}}{2 \lambda }+\frac {1}{c_{1} +\lambda \sqrt {\pi }\, 2^{n} \left (\frac {\arccos \left (x \right )^{1+n} 2^{-n} \sqrt {-x^{2}+1}}{\sqrt {\pi }\, \left (n +2\right )}-\frac {2^{-n} \sqrt {\arccos \left (x \right )}\, \operatorname {LommelS1}\left (\frac {3}{2}+n , \frac {3}{2}, \arccos \left (x \right )\right ) \sqrt {-x^{2}+1}}{\sqrt {\pi }\, \left (n +2\right )}-\frac {3 \,2^{-1-n} \left (\frac {2 n}{3}+\frac {4}{3}\right ) \left (\arccos \left (x \right ) x -\sqrt {-x^{2}+1}\right ) \operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (x \right )\right )}{\sqrt {\pi }\, \left (n +2\right ) \sqrt {\arccos \left (x \right )}}\right )} \]
✓ Solution by Mathematica
Time used: 4.776 (sec). Leaf size: 86
DSolve[y'[x]==\[Lambda]*ArcCos[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} \lambda \arccos (x)^n (-i \arccos (x))^{-n} \Gamma (n+1,-i \arccos (x))-\frac {1}{2} \lambda (i \arccos (x))^{-n} \arccos (x)^n \Gamma (n+1,i \arccos (x))+c_1}+b y(x)\to a x^m+b \end{align*}