10.14.11 problem 7. case \(x_0=2\)

Internal problem ID [1393]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 5.3, Series Solutions Near an Ordinary Point, Part II. page 269
Problem number : 7. case \(x_0=2\)
Date solved : Saturday, March 29, 2025 at 10:54:05 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{3}+1\right ) y^{\prime \prime }+4 x y^{\prime }+y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 2 \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 76

Order:=6; 
ode:=(x^3+1)*diff(diff(y(x),x),x)+4*x*diff(y(x),x)+y(x) = 0; 
dsolve(ode,y(x),type='series',x=2);
 

\[ y = \left (1-\frac {\left (x -2\right )^{2}}{18}+\frac {10 \left (x -2\right )^{3}}{243}-\frac {451 \left (x -2\right )^{4}}{17496}+\frac {1151 \left (x -2\right )^{5}}{78732}\right ) y \left (2\right )+\left (x -2-\frac {4 \left (x -2\right )^{2}}{9}+\frac {115 \left (x -2\right )^{3}}{486}-\frac {271 \left (x -2\right )^{4}}{2187}+\frac {9713 \left (x -2\right )^{5}}{157464}\right ) y^{\prime }\left (2\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 87

ode=(1+x^3)*D[y[x],{x,2}]+4*x*D[y[x],x]+y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,2,5}]
 

\[ y(x)\to c_1 \left (\frac {1151 (x-2)^5}{78732}-\frac {451 (x-2)^4}{17496}+\frac {10}{243} (x-2)^3-\frac {1}{18} (x-2)^2+1\right )+c_2 \left (\frac {9713 (x-2)^5}{157464}-\frac {271 (x-2)^4}{2187}+\frac {115}{486} (x-2)^3-\frac {4}{9} (x-2)^2+x-2\right ) \]

✓ Sympy. Time used: 1.852 (sec). Leaf size: 63

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*x*Derivative(y(x), x) + (x**3 + 1)*Derivative(y(x), (x, 2)) + y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=2,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (x - \frac {271 \left (x - 2\right )^{4}}{2187} + \frac {115 \left (x - 2\right )^{3}}{486} - \frac {4 \left (x - 2\right )^{2}}{9} - 2\right ) + C_{1} \left (- \frac {451 \left (x - 2\right )^{4}}{17496} + \frac {10 \left (x - 2\right )^{3}}{243} - \frac {\left (x - 2\right )^{2}}{18} + 1\right ) + O\left (x^{6}\right ) \]