Internal problem ID [9830]
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]
Solve \begin {gather*} \boxed {y^{\prime }-\lambda \arccos \relax (x )^{n} \left (y-a \,x^{m}-b \right )^{2}-a m \,x^{m -1}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (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)
\[ y \relax (x ) = -\frac {\left (-2 \arccos \relax (x )^{n} x^{m} a \lambda -2 \arccos \relax (x )^{n} \lambda b \right ) \arccos \relax (x )^{-n}}{2 \lambda }+\frac {1}{c_{1}+\lambda 2^{n} \sqrt {\pi }\, \left (\frac {\arccos \relax (x )^{n +1} 2^{-n} \sqrt {-x^{2}+1}}{\sqrt {\pi }\, \left (2+n \right )}-\frac {2^{-n} \sqrt {\arccos \relax (x )}\, \LommelS 1 \left (n +\frac {3}{2}, \frac {3}{2}, \arccos \relax (x )\right ) \sqrt {-x^{2}+1}}{\sqrt {\pi }\, \left (2+n \right )}-\frac {3 \,2^{-n -1} \left (\frac {4}{3}+\frac {2 n}{3}\right ) \left (x \arccos \relax (x )-\sqrt {-x^{2}+1}\right ) \LommelS 1 \left (n +\frac {1}{2}, \frac {1}{2}, \arccos \relax (x )\right )}{\sqrt {\pi }\, \left (2+n \right ) \sqrt {\arccos \relax (x )}}\right )} \]
✓ Solution by Mathematica
Time used: 3.791 (sec). Leaf size: 84
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}{c_1-\frac {1}{2} \lambda \text {ArcCos}(x)^n \left (\text {ArcCos}(x)^2\right )^{-n} \left ((-i \text {ArcCos}(x))^n \Gamma (n+1,i \text {ArcCos}(x))+(i \text {ArcCos}(x))^n \Gamma (n+1,-i \text {ArcCos}(x))\right )}+b \\ y(x)\to a x^m+b \\ \end{align*}