61.15.1 problem 10

Internal problem ID [12164]
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
Date solved : Friday, March 14, 2025 at 04:31:59 AM
CAS classification : [_Riccati]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 384

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

\[ \text {Expression too large to display} \]

✓ Mathematica. Time used: 1.044 (sec). Leaf size: 220

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

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

✓ Sympy. Time used: 8.657 (sec). Leaf size: 76

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

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