64.15.6 problem 6

Internal problem ID [13512]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 6, Series solutions of linear differential equations. Section 6.2 (Frobenius). Exercises page 251
Problem number : 6
Date solved : Monday, March 31, 2025 at 08:00:01 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 x^{2} y^{\prime \prime }+x y^{\prime }+\left (2 x^{2}-3\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.022 (sec). Leaf size: 36

Order:=6; 
ode:=2*x^2*diff(diff(y(x),x),x)+x*diff(y(x),x)+(2*x^2-3)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_2 \,x^{{5}/{2}} \left (1-\frac {1}{9} x^{2}+\frac {1}{234} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_1 \left (1+x^{2}-\frac {1}{6} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 46

ode=2*x^2*D[y[x],{x,2}]+x*D[y[x],x]+(2*x^2-3)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to \frac {c_2 \left (-\frac {x^4}{6}+x^2+1\right )}{x}+c_1 \left (\frac {x^4}{234}-\frac {x^2}{9}+1\right ) x^{3/2} \]

✓ Sympy. Time used: 0.879 (sec). Leaf size: 37

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x**2*Derivative(y(x), (x, 2)) + x*Derivative(y(x), x) + (2*x**2 - 3)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} x^{\frac {3}{2}} \left (1 - \frac {x^{2}}{9}\right ) + \frac {C_{1} \left (\frac {x^{6}}{126} - \frac {x^{4}}{6} + x^{2} + 1\right )}{x} + O\left (x^{6}\right ) \]