Internal problem ID [9815]
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: 1.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {y^{\prime }-y^{2}-\lambda \arcsin \relax (x )^{n} y+a^{2}-a \lambda \arcsin \relax (x )^{n}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.002 (sec). Leaf size: 97
dsolve(diff(y(x),x)=y(x)^2+lambda*arcsin(x)^n*y(x)-a^2+a*lambda*arcsin(x)^n,y(x), singsol=all)
\[ y \relax (x ) = -\frac {\left (\left (\int {\mathrm e}^{\int \left (\arcsin \relax (x )^{n} \lambda -2 a \right )d x}d x \right ) {\mathrm e}^{\int \left (-\arcsin \relax (x )^{n} \lambda +2 a \right )d x} a +c_{1} {\mathrm e}^{\int \left (-\arcsin \relax (x )^{n} \lambda +2 a \right )d x} a +1\right ) {\mathrm e}^{\int \left (\arcsin \relax (x )^{n} \lambda -2 a \right )d x}}{c_{1}+\int {\mathrm e}^{\int \left (\arcsin \relax (x )^{n} \lambda -2 a \right )d x}d x} \]
✓ Solution by Mathematica
Time used: 3.067 (sec). Leaf size: 398
DSolve[y'[x]==y[x]^2+\[Lambda]*ArcSin[x]^n*y[x]-a^2+a*\[Lambda]*ArcSin[x]^n,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^x-\frac {\exp \left (\frac {1}{2} i \lambda \text {ArcSin}(K[1])^n \left (\text {ArcSin}(K[1])^2\right )^{-n} \left ((-i \text {ArcSin}(K[1]))^n \text {Gamma}(n+1,i \text {ArcSin}(K[1]))-(i \text {ArcSin}(K[1]))^n \text {Gamma}(n+1,-i \text {ArcSin}(K[1]))\right )-2 a K[1]\right ) \left (\lambda \text {ArcSin}(K[1])^n-a+y(x)\right )}{n \lambda (a+y(x))}dK[1]+\int _1^{y(x)}\left (\frac {\exp \left (\frac {1}{2} i \lambda \text {ArcSin}(x)^n \left (\text {ArcSin}(x)^2\right )^{-n} \left ((-i \text {ArcSin}(x))^n \text {Gamma}(n+1,i \text {ArcSin}(x))-(i \text {ArcSin}(x))^n \text {Gamma}(n+1,-i \text {ArcSin}(x))\right )-2 a x\right )}{n \lambda (a+K[2])^2}-\int _1^x\left (\frac {\exp \left (\frac {1}{2} i \lambda \text {ArcSin}(K[1])^n \left (\text {ArcSin}(K[1])^2\right )^{-n} \left ((-i \text {ArcSin}(K[1]))^n \text {Gamma}(n+1,i \text {ArcSin}(K[1]))-(i \text {ArcSin}(K[1]))^n \text {Gamma}(n+1,-i \text {ArcSin}(K[1]))\right )-2 a K[1]\right ) \left (\lambda \text {ArcSin}(K[1])^n-a+K[2]\right )}{n \lambda (a+K[2])^2}-\frac {\exp \left (\frac {1}{2} i \lambda \text {ArcSin}(K[1])^n \left (\text {ArcSin}(K[1])^2\right )^{-n} \left ((-i \text {ArcSin}(K[1]))^n \text {Gamma}(n+1,i \text {ArcSin}(K[1]))-(i \text {ArcSin}(K[1]))^n \text {Gamma}(n+1,-i \text {ArcSin}(K[1]))\right )-2 a K[1]\right )}{n \lambda (a+K[2])}\right )dK[1]\right )dK[2]=c_1,y(x)\right ] \]