55.14.1 problem 1

Internal problem ID [13454]
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
Date solved : Wednesday, October 01, 2025 at 01:46:29 PM
CAS classification : [_Riccati]

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 71

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

\[ y = \frac {-\int {\mathrm e}^{-\int \left (-\lambda \arcsin \left (x \right )^{n}+2 a \right )d x}d x a -c_1 a -{\mathrm e}^{-\int \left (-\lambda \arcsin \left (x \right )^{n}+2 a \right )d x}}{c_1 +\int {\mathrm e}^{-\int \left (-\lambda \arcsin \left (x \right )^{n}+2 a \right )d x}d x} \]

✓ Mathematica. Time used: 0.727 (sec). Leaf size: 210

ode=D[y[x],x]==y[x]^2+\[Lambda]*ArcSin[x]^n*y[x]-a^2+a*\[Lambda]*ArcSin[x]^n; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 5.708 (sec). Leaf size: 76

from sympy import * 
x = symbols("x") 
a = symbols("a") 
lambda_ = symbols("lambda_") 
n = symbols("n") 
y = Function("y") 
ode = Eq(a**2 - a*lambda_*asin(x)**n - lambda_*y(x)*asin(x)**n - y(x)**2 + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {\left (- C_{1} a e^{2 a x} + a e^{2 a x} \int e^{- 2 a x} e^{\lambda _{} \int \operatorname {asin}^{n}{\left (x \right )}\, dx}\, dx + e^{\lambda _{} \int \operatorname {asin}^{n}{\left (x \right )}\, dx}\right ) e^{- 2 a x}}{C_{1} - \int e^{- 2 a x} e^{\lambda _{} \int \operatorname {asin}^{n}{\left (x \right )}\, dx}\, dx} \]