61.14.4 problem 4

Internal problem ID [12237]
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 : 4
Date solved : Tuesday, January 28, 2025 at 07:48:22 PM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&=\lambda \arcsin \left (x \right )^{n} y^{2}+a y+a b -b^{2} \lambda \arcsin \left (x \right )^{n} \end{align*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 87

dsolve(diff(y(x),x)=lambda*arcsin(x)^n*y(x)^2+a*y(x)+a*b-b^2*lambda*arcsin(x)^n,y(x), singsol=all)
 

\[ y = \frac {-b \lambda \left (\int \arcsin \left (x \right )^{n} {\mathrm e}^{-\int \left (2 \arcsin \left (x \right )^{n} \lambda b -a \right )d x}d x \right )-c_{1} b -{\mathrm e}^{-\int \left (2 \arcsin \left (x \right )^{n} \lambda b -a \right )d x}}{c_{1} +\lambda \left (\int \arcsin \left (x \right )^{n} {\mathrm e}^{-\int \left (2 \arcsin \left (x \right )^{n} \lambda b -a \right )d x}d x \right )} \]

✓ Solution by Mathematica

Time used: 1.585 (sec). Leaf size: 250

DSolve[D[y[x],x]==\[Lambda]*ArcSin[x]^n*y[x]^2+a*y[x]+a*b-b^2*\[Lambda]*ArcSin[x]^n,y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\int _1^x\frac {i \exp \left (-\int _1^{K[2]}\left (2 b \lambda \arcsin (K[1])^n-a\right )dK[1]\right ) \left (-b \lambda \arcsin (K[2])^n+\lambda y(x) \arcsin (K[2])^n+a\right )}{a n \lambda (b+y(x))}dK[2]+\int _1^{y(x)}\left (-\int _1^x\left (\frac {i \exp \left (-\int _1^{K[2]}\left (2 b \lambda \arcsin (K[1])^n-a\right )dK[1]\right ) \arcsin (K[2])^n}{a n (b+K[3])}-\frac {i \exp \left (-\int _1^{K[2]}\left (2 b \lambda \arcsin (K[1])^n-a\right )dK[1]\right ) \left (-b \lambda \arcsin (K[2])^n+\lambda K[3] \arcsin (K[2])^n+a\right )}{a n \lambda (b+K[3])^2}\right )dK[2]-\frac {i \exp \left (-\int _1^x\left (2 b \lambda \arcsin (K[1])^n-a\right )dK[1]\right )}{a n \lambda (b+K[3])^2}\right )dK[3]=c_1,y(x)\right ] \]