44.17.20 problem 3

Internal problem ID [9377]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 4. Power Series Solutions and Special Functions. Section 4.2. Series Solutions of First-Order Differential Equations Page 162
Problem number : 3
Date solved : Tuesday, September 30, 2025 at 06:17:59 PM
CAS classification : [_quadrature]

\begin{align*} y^{\prime }&=\frac {1}{\sqrt {-x^{2}+1}} \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 27

Order:=8; 
ode:=diff(y(x),x) = 1/(-x^2+1)^(1/2); 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = y \left (0\right )+x +\frac {x^{3}}{6}+\frac {3 x^{5}}{40}+\frac {5 x^{7}}{112}+O\left (x^{8}\right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 28

ode=D[y[x],x]==(1-x^2)^(-1/2); 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to \frac {5 x^7}{112}+\frac {3 x^5}{40}+\frac {x^3}{6}+x+c_1 \]

✓ Sympy. Time used: 0.230 (sec). Leaf size: 27

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) - 1/sqrt(1 - x**2),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="1st_power_series",x0=0,n=8)
 

\[ y{\left (x \right )} = x + \frac {x^{3}}{6} + \frac {3 x^{5}}{40} + \frac {5 x^{7}}{112} + C_{1} + O\left (x^{8}\right ) \]