Internal problem ID [9824]
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: 10.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {y^{\prime }-y^{2}-\lambda \arccos \relax (x )^{n} y+a^{2}-a \lambda \arccos \relax (x )^{n}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.002 (sec). Leaf size: 595
dsolve(diff(y(x),x)=y(x)^2+lambda*arccos(x)^n*y(x)-a^2+a*lambda*arccos(x)^n,y(x), singsol=all)
\[ \text {Expression too large to display} \]
✓ Solution by Mathematica
Time used: 3.543 (sec). Leaf size: 404
DSolve[y'[x]==y[x]^2+\[Lambda]*ArcCos[x]^n*y[x]-a^2+a*\[Lambda]*ArcCos[x]^n,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^x\frac {i \exp \left (\frac {1}{2} \lambda \text {ArcCos}(K[1])^n \text {Gamma}(n+1,-i \text {ArcCos}(K[1])) (-i \text {ArcCos}(K[1]))^{-n}+\frac {1}{2} \lambda (i \text {ArcCos}(K[1]))^{-n} \text {ArcCos}(K[1])^n \text {Gamma}(n+1,i \text {ArcCos}(K[1]))-2 a K[1]\right ) \left (\lambda \text {ArcCos}(K[1])^n-a+y(x)\right )}{n \lambda (a+y(x))}dK[1]+\int _1^{y(x)}\left (-\int _1^x\left (\frac {i \exp \left (\frac {1}{2} \lambda \text {ArcCos}(K[1])^n \text {Gamma}(n+1,-i \text {ArcCos}(K[1])) (-i \text {ArcCos}(K[1]))^{-n}+\frac {1}{2} \lambda (i \text {ArcCos}(K[1]))^{-n} \text {ArcCos}(K[1])^n \text {Gamma}(n+1,i \text {ArcCos}(K[1]))-2 a K[1]\right )}{n \lambda (a+K[2])}-\frac {i \exp \left (\frac {1}{2} \lambda \text {ArcCos}(K[1])^n \text {Gamma}(n+1,-i \text {ArcCos}(K[1])) (-i \text {ArcCos}(K[1]))^{-n}+\frac {1}{2} \lambda (i \text {ArcCos}(K[1]))^{-n} \text {ArcCos}(K[1])^n \text {Gamma}(n+1,i \text {ArcCos}(K[1]))-2 a K[1]\right ) \left (\lambda \text {ArcCos}(K[1])^n-a+K[2]\right )}{n \lambda (a+K[2])^2}\right )dK[1]-\frac {i \exp \left (\frac {1}{2} \lambda \text {ArcCos}(x)^n \text {Gamma}(n+1,-i \text {ArcCos}(x)) (-i \text {ArcCos}(x))^{-n}-2 a x+\frac {1}{2} \lambda (i \text {ArcCos}(x))^{-n} \text {ArcCos}(x)^n \text {Gamma}(n+1,i \text {ArcCos}(x))\right )}{n \lambda (a+K[2])^2}\right )dK[2]=c_1,y(x)\right ] \]