7.15.29 problem 29

Internal problem ID [485]
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 : 29
Date solved : Saturday, March 29, 2025 at 04:54:46 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x y^{\prime \prime }+8 y^{\prime }+x y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.023 (sec). Leaf size: 32

Order:=6; 
ode:=4*x*diff(diff(y(x),x),x)+8*diff(y(x),x)+x*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \left (1-\frac {1}{24} x^{2}+\frac {1}{1920} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_2 \left (1-\frac {1}{8} x^{2}+\frac {1}{384} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

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

ode=4*x*D[y[x],{x,2}]+8*D[y[x],x]+x*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {x^3}{384}-\frac {x}{8}+\frac {1}{x}\right )+c_2 \left (\frac {x^4}{1920}-\frac {x^2}{24}+1\right ) \]

✓ Sympy. Time used: 0.859 (sec). Leaf size: 39

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*y(x) + 4*x*Derivative(y(x), (x, 2)) + 8*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{4}}{1920} - \frac {x^{2}}{24} + 1\right ) + \frac {C_{1} \left (- \frac {x^{6}}{46080} + \frac {x^{4}}{384} - \frac {x^{2}}{8} + 1\right )}{x} + O\left (x^{6}\right ) \]