7.15.31 problem 31

Internal problem ID [487]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 3. Power series methods. Section 3.3 (Regular singular points). Problems at page 231
Problem number : 31
Date solved : Tuesday, March 04, 2025 at 11:25:10 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} y^{\prime \prime }-4 x y^{\prime }+\left (-4 x^{2}+3\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 34

Order:=6; 
ode:=4*x^2*diff(diff(y(x),x),x)-4*x*diff(y(x),x)+(-4*x^2+3)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \sqrt {x}\, \left (\left (1+\frac {1}{6} x^{2}+\frac {1}{120} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) x c_1 +\left (1+\frac {1}{2} x^{2}+\frac {1}{24} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \right ) \]

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 58

ode=4*x^2*D[y[x],{x,2}]-4*x*D[y[x],x]+(3-4*x^2)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {x^{9/2}}{24}+\frac {x^{5/2}}{2}+\sqrt {x}\right )+c_2 \left (\frac {x^{11/2}}{120}+\frac {x^{7/2}}{6}+x^{3/2}\right ) \]

✓ Sympy. Time used: 0.950 (sec). Leaf size: 37

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*x**2*Derivative(y(x), (x, 2)) - 4*x*Derivative(y(x), x) + (3 - 4*x**2)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} x^{\frac {3}{2}} \left (\frac {x^{2}}{6} + 1\right ) + C_{1} \sqrt {x} \left (\frac {x^{4}}{24} + \frac {x^{2}}{2} + 1\right ) + O\left (x^{6}\right ) \]