7.15.27 problem 27

Internal problem ID [483]
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 : 27
Date solved : Saturday, March 29, 2025 at 04:54:43 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x y^{\prime \prime }+2 y^{\prime }+9 x y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.028 (sec). Leaf size: 32

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

\[ y = c_1 \left (1-\frac {3}{2} x^{2}+\frac {27}{40} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_2 \left (1-\frac {9}{2} x^{2}+\frac {27}{8} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 42

ode=x*D[y[x],{x,2}]+2*D[y[x],x]+9*x*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {27 x^3}{8}-\frac {9 x}{2}+\frac {1}{x}\right )+c_2 \left (\frac {27 x^4}{40}-\frac {3 x^2}{2}+1\right ) \]

✓ Sympy. Time used: 0.827 (sec). Leaf size: 48

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

\[ y{\left (x \right )} = C_{2} \left (\frac {27 x^{4}}{40} - \frac {3 x^{2}}{2} + 1\right ) + \frac {C_{1} \left (- \frac {81 x^{6}}{80} + \frac {27 x^{4}}{8} - \frac {9 x^{2}}{2} + 1\right )}{x} + O\left (x^{6}\right ) \]