10.14.9 problem 6. case \(x_0=-4\)

Internal problem ID [1391]
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 : 6. case \(x_0=-4\)
Date solved : Tuesday, March 04, 2025 at 12:35:03 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{2}-2 x -3\right ) y^{\prime \prime }+x y^{\prime }+4 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 76

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

\[ y = \left (1-\frac {2 \left (x +4\right )^{2}}{21}-\frac {4 \left (x +4\right )^{3}}{189}-\frac {4 \left (x +4\right )^{4}}{1323}-\frac {\left (x +4\right )^{5}}{3087}\right ) y \left (-4\right )+\left (x +4+\frac {2 \left (x +4\right )^{2}}{21}-\frac {\left (x +4\right )^{3}}{54}-\frac {11 \left (x +4\right )^{4}}{1323}-\frac {157 \left (x +4\right )^{5}}{74088}\right ) y^{\prime }\left (-4\right )+O\left (x^{6}\right ) \]

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

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

\[ y(x)\to c_1 \left (-\frac {(x+4)^5}{3087}-\frac {4 (x+4)^4}{1323}-\frac {4}{189} (x+4)^3-\frac {2}{21} (x+4)^2+1\right )+c_2 \left (-\frac {157 (x+4)^5}{74088}-\frac {11 (x+4)^4}{1323}-\frac {1}{54} (x+4)^3+\frac {2}{21} (x+4)^2+x+4\right ) \]

✓ Sympy. Time used: 0.913 (sec). Leaf size: 63

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

\[ y{\left (x \right )} = C_{2} \left (x - \frac {11 \left (x + 4\right )^{4}}{1323} - \frac {\left (x + 4\right )^{3}}{54} + \frac {2 \left (x + 4\right )^{2}}{21} + 4\right ) + C_{1} \left (- \frac {4 \left (x + 4\right )^{4}}{1323} - \frac {4 \left (x + 4\right )^{3}}{189} - \frac {2 \left (x + 4\right )^{2}}{21} + 1\right ) + O\left (x^{6}\right ) \]