10.14.8 problem 6. case \(x_0=4\) only

Internal problem ID [1390]
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 \(x_0=4\) only
Date solved : Tuesday, March 04, 2025 at 12:35:02 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: 0.003 (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);
 

\[ y = \left (1-\frac {2 \left (x -4\right )^{2}}{5}+\frac {4 \left (x -4\right )^{3}}{15}-\frac {4 \left (x -4\right )^{4}}{25}+\frac {199 \left (x -4\right )^{5}}{1875}\right ) y \left (4\right )+\left (x -4-\frac {2 \left (x -4\right )^{2}}{5}+\frac {\left (x -4\right )^{3}}{10}-\frac {2 \left (x -4\right )^{4}}{75}+\frac {157 \left (x -4\right )^{5}}{15000}\right ) y^{\prime }\left (4\right )+O\left (x^{6}\right ) \]

✓ 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}]
 

\[ y(x)\to c_1 \left (\frac {199 (x-4)^5}{1875}-\frac {4}{25} (x-4)^4+\frac {4}{15} (x-4)^3-\frac {2}{5} (x-4)^2+1\right )+c_2 \left (\frac {157 (x-4)^5}{15000}-\frac {2}{75} (x-4)^4+\frac {1}{10} (x-4)^3-\frac {2}{5} (x-4)^2+x-4\right ) \]

✓ Sympy. Time used: 0.867 (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)
 

\[ y{\left (x \right )} = C_{2} \left (x - \frac {2 \left (x - 4\right )^{4}}{75} + \frac {\left (x - 4\right )^{3}}{10} - \frac {2 \left (x - 4\right )^{2}}{5} - 4\right ) + C_{1} \left (- \frac {4 \left (x - 4\right )^{4}}{25} + \frac {4 \left (x - 4\right )^{3}}{15} - \frac {2 \left (x - 4\right )^{2}}{5} + 1\right ) + O\left (x^{6}\right ) \]