40.17.3 problem 13

Internal problem ID [6804]
Book : Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres. McGraw Hill 1952
Section : Chapter 26. Integration in series (singular points). Supplemetary problems. Page 218
Problem number : 13
Date solved : Sunday, March 30, 2025 at 11:23:15 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.016 (sec). Leaf size: 33

Order:=6; 
ode:=2*x^2*diff(diff(y(x),x),x)-x*diff(y(x),x)+(-x^2+1)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \sqrt {x}\, \left (1+\frac {1}{6} x^{2}+\frac {1}{168} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_2 x \left (1+\frac {1}{10} x^{2}+\frac {1}{360} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 48

ode=2*x^2*D[y[x],{x,2}]-x*D[y[x],x]+(1-x^2)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 x \left (\frac {x^4}{360}+\frac {x^2}{10}+1\right )+c_2 \sqrt {x} \left (\frac {x^4}{168}+\frac {x^2}{6}+1\right ) \]

✓ Sympy. Time used: 1.091 (sec). Leaf size: 39

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x**2*Derivative(y(x), (x, 2)) - x*Derivative(y(x), x) + (1 - x**2)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} x \left (\frac {x^{4}}{360} + \frac {x^{2}}{10} + 1\right ) + C_{1} \sqrt {x} \left (\frac {x^{4}}{168} + \frac {x^{2}}{6} + 1\right ) + O\left (x^{6}\right ) \]