50.17.20 problem 3

Internal problem ID [8091]
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 : Monday, January 27, 2025 at 03:43:35 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*}

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 27

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

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

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 28

AsymptoticDSolveValue[D[y[x],x]==(1-x^2)^(-1/2),y[x],{x,0,"8"-1}]
 

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