Internal problem ID [9827]
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: 13.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {y^{\prime }-\lambda \arccos \relax (x )^{n} y^{2}-a y-a b +b^{2} \lambda \arccos \relax (x )^{n}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.009 (sec). Leaf size: 587
dsolve(diff(y(x),x)=lambda*arccos(x)^n*y(x)^2+a*y(x)+a*b-b^2*lambda*arccos(x)^n,y(x), singsol=all)
\[ \text {Expression too large to display} \]
✓ Solution by Mathematica
Time used: 4.99 (sec). Leaf size: 420
DSolve[y'[x]==\[Lambda]*ArcCos[x]^n*y[x]^2+a*y[x]+a*b-b^2*\[Lambda]*ArcCos[x]^n,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^x-\frac {i \exp \left (-b \lambda \text {ArcCos}(K[1])^n \text {Gamma}(n+1,-i \text {ArcCos}(K[1])) (-i \text {ArcCos}(K[1]))^{-n}-b \lambda (i \text {ArcCos}(K[1]))^{-n} \text {ArcCos}(K[1])^n \text {Gamma}(n+1,i \text {ArcCos}(K[1]))+a K[1]\right ) \left (-b \lambda \text {ArcCos}(K[1])^n+\lambda y(x) \text {ArcCos}(K[1])^n+a\right )}{a n \lambda (b+y(x))}dK[1]+\int _1^{y(x)}\left (\frac {i \exp \left (-b \lambda \text {ArcCos}(x)^n \text {Gamma}(n+1,-i \text {ArcCos}(x)) (-i \text {ArcCos}(x))^{-n}+a x-b \lambda (i \text {ArcCos}(x))^{-n} \text {ArcCos}(x)^n \text {Gamma}(n+1,i \text {ArcCos}(x))\right )}{a n \lambda (b+K[2])^2}-\int _1^x\left (\frac {i \exp \left (-b \lambda \text {ArcCos}(K[1])^n \text {Gamma}(n+1,-i \text {ArcCos}(K[1])) (-i \text {ArcCos}(K[1]))^{-n}-b \lambda (i \text {ArcCos}(K[1]))^{-n} \text {ArcCos}(K[1])^n \text {Gamma}(n+1,i \text {ArcCos}(K[1]))+a K[1]\right ) \left (-b \lambda \text {ArcCos}(K[1])^n+\lambda K[2] \text {ArcCos}(K[1])^n+a\right )}{a n \lambda (b+K[2])^2}-\frac {i \exp \left (-b \lambda \text {ArcCos}(K[1])^n \text {Gamma}(n+1,-i \text {ArcCos}(K[1])) (-i \text {ArcCos}(K[1]))^{-n}-b \lambda (i \text {ArcCos}(K[1]))^{-n} \text {ArcCos}(K[1])^n \text {Gamma}(n+1,i \text {ArcCos}(K[1]))+a K[1]\right ) \text {ArcCos}(K[1])^n}{a n (b+K[2])}\right )dK[1]\right )dK[2]=c_1,y(x)\right ] \]