10.14.6 problem 5. case \(x_0=4\)

Internal problem ID [1388]
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 : 5. case \(x_0=4\)
Date solved : Saturday, March 29, 2025 at 10:53:58 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+6 x y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 4 \end{align*}

✓ Maple. Time used: 0.009 (sec). Leaf size: 76

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

\[ y = \left (1-12 \left (x -4\right )^{2}+15 \left (x -4\right )^{3}+9 \left (x -4\right )^{4}-\frac {108 \left (x -4\right )^{5}}{5}\right ) y \left (4\right )+\left (x -4-2 \left (x -4\right )^{2}-\frac {4 \left (x -4\right )^{3}}{3}+\frac {29 \left (x -4\right )^{4}}{6}-\frac {5 \left (x -4\right )^{5}}{3}\right ) y^{\prime }\left (4\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 79

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

\[ y(x)\to c_1 \left (-\frac {108}{5} (x-4)^5+9 (x-4)^4+15 (x-4)^3-12 (x-4)^2+1\right )+c_2 \left (-\frac {5}{3} (x-4)^5+\frac {29}{6} (x-4)^4-\frac {4}{3} (x-4)^3-2 (x-4)^2+x-4\right ) \]

✓ Sympy. Time used: 0.715 (sec). Leaf size: 58

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

\[ y{\left (x \right )} = C_{2} \left (x + \frac {29 \left (x - 4\right )^{4}}{6} - \frac {4 \left (x - 4\right )^{3}}{3} - 2 \left (x - 4\right )^{2} - 4\right ) + C_{1} \left (9 \left (x - 4\right )^{4} + 15 \left (x - 4\right )^{3} - 12 \left (x - 4\right )^{2} + 1\right ) + O\left (x^{6}\right ) \]