46.2.11 problem 12

Internal problem ID [7314]
Book : ADVANCED ENGINEERING MATHEMATICS. ERWIN KREYSZIG, HERBERT KREYSZIG, EDWARD J. NORMINTON. 10th edition. John Wiley USA. 2011
Section : Chapter 5. Series Solutions of ODEs. Special Functions. Problem set 5.3. Extended Power Series Method: Frobenius Method page 186
Problem number : 12
Date solved : Wednesday, March 05, 2025 at 04:23:09 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 34

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

\[ y = \frac {c_{1} \left (1-\frac {2}{3} x^{2}+\frac {2}{15} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) x +c_{2} \left (1-2 x^{2}+\frac {2}{3} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{3}} \]

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 38

ode=x^2*D[y[x],{x,2}]+6*x*D[y[x],x]+(4*x^2+6)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {1}{x^3}+\frac {2 x}{3}-\frac {2}{x}\right )+c_2 \left (\frac {2 x^2}{15}+\frac {1}{x^2}-\frac {2}{3}\right ) \]

✓ Sympy. Time used: 1.020 (sec). Leaf size: 36

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

\[ y{\left (x \right )} = \frac {C_{1} \left (\frac {2 x^{8}}{315} - \frac {4 x^{6}}{45} + \frac {2 x^{4}}{3} - 2 x^{2} + 1\right )}{x^{3}} + O\left (1\right ) \]