54.5.13 problem 12

Internal problem ID [8628]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 18. Power series solutions. 18.6. Indicial Equation with Equal Roots. Exercises page 373
Problem number : 12
Date solved : Sunday, March 30, 2025 at 01:21:41 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x y^{\prime \prime }+\left (-x^{2}+1\right ) y^{\prime }-x y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.026 (sec). Leaf size: 34

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

\[ y = \left (c_2 \ln \left (x \right )+c_1 \right ) \left (1+\frac {1}{4} x^{2}+\frac {3}{64} x^{4}+\frac {5}{768} x^{6}+\operatorname {O}\left (x^{8}\right )\right )+\left (-\frac {1}{128} x^{4}-\frac {1}{512} x^{6}+\operatorname {O}\left (x^{8}\right )\right ) c_2 \]

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 74

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

\[ y(x)\to c_1 \left (\frac {5 x^6}{768}+\frac {3 x^4}{64}+\frac {x^2}{4}+1\right )+c_2 \left (-\frac {x^6}{512}-\frac {x^4}{128}+\left (\frac {5 x^6}{768}+\frac {3 x^4}{64}+\frac {x^2}{4}+1\right ) \log (x)\right ) \]

✓ Sympy. Time used: 0.911 (sec). Leaf size: 27

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

\[ y{\left (x \right )} = C_{1} \left (\frac {5 x^{6}}{768} + \frac {3 x^{4}}{64} + \frac {x^{2}}{4} + 1\right ) + O\left (x^{8}\right ) \]