7.16.15 problem 16

Internal problem ID [512]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 3. Power series methods. Section 3.4 (Method of Frobenius: The exceptional cases). Problems at page 246
Problem number : 16
Date solved : Tuesday, March 04, 2025 at 11:25:43 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {9}{4}\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 36

Order:=6; 
ode:=x^2*diff(diff(y(x),x),x)+x*diff(y(x),x)+(x^2-9/4)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_1 \,x^{3} \left (1-\frac {1}{10} x^{2}+\frac {1}{280} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (12+6 x^{2}-\frac {3}{2} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{{3}/{2}}} \]

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

ode=x^2*D[y[x],{x,2}]+x*D[y[x],x]+(x^2-9/4)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (-\frac {x^{5/2}}{8}+\frac {1}{x^{3/2}}+\frac {\sqrt {x}}{2}\right )+c_2 \left (\frac {x^{11/2}}{280}-\frac {x^{7/2}}{10}+x^{3/2}\right ) \]

✓ Sympy. Time used: 0.899 (sec). Leaf size: 42

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + x*Derivative(y(x), x) + (x**2 - 9/4)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} x^{\frac {3}{2}} \left (1 - \frac {x^{2}}{10}\right ) + \frac {C_{1} \left (\frac {x^{6}}{144} - \frac {x^{4}}{8} + \frac {x^{2}}{2} + 1\right )}{x^{\frac {3}{2}}} + O\left (x^{6}\right ) \]