54.3.11 problem 11

Internal problem ID [8567]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 17. Power series solutions. 17.5. Solutions Near an Ordinary Point. Exercises page 355
Problem number : 11
Date solved : Wednesday, March 05, 2025 at 06:09:44 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} \left (x^{2}+4\right ) y^{\prime \prime }+6 x y^{\prime }+4 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 49

Order:=8; 
ode:=(x^2+4)*diff(diff(y(x),x),x)+6*x*diff(y(x),x)+4*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1-\frac {1}{2} x^{2}+\frac {3}{16} x^{4}-\frac {1}{16} x^{6}\right ) y \left (0\right )+\left (x -\frac {5}{12} x^{3}+\frac {7}{48} x^{5}-\frac {3}{64} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 56

ode=(x^2+4)*D[y[x],{x,2}]+6*x*D[y[x],x]+4*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_2 \left (-\frac {3 x^7}{64}+\frac {7 x^5}{48}-\frac {5 x^3}{12}+x\right )+c_1 \left (-\frac {x^6}{16}+\frac {3 x^4}{16}-\frac {x^2}{2}+1\right ) \]

✓ Sympy. Time used: 0.894 (sec). Leaf size: 44

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

\[ y{\left (x \right )} = C_{2} \left (- \frac {x^{6}}{16} + \frac {3 x^{4}}{16} - \frac {x^{2}}{2} + 1\right ) + C_{1} x \left (\frac {7 x^{4}}{48} - \frac {5 x^{2}}{12} + 1\right ) + O\left (x^{8}\right ) \]