55.15.4 problem 13

Internal problem ID [13466]
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
Date solved : Wednesday, October 01, 2025 at 02:21:33 PM
CAS classification : [_Riccati]

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 374

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

\begin{align*} \text {Solution too large to show}\end{align*}

✓ Mathematica. Time used: 1.34 (sec). Leaf size: 250

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

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

✓ Sympy. Time used: 39.104 (sec). Leaf size: 109

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

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