10.14.7 problem 6. case \(x_0=0\)

Internal problem ID [1389]
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 \(x_0=0\)
Date solved : Saturday, March 29, 2025 at 10:53:59 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*} 0 \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 54

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=0);
 

\[ y = \left (1+\frac {2}{3} x^{2}-\frac {4}{27} x^{3}+\frac {16}{81} x^{4}-\frac {1}{9} x^{5}\right ) y \left (0\right )+\left (x +\frac {5}{18} x^{3}-\frac {5}{54} x^{4}+\frac {7}{72} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 63

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,0,5}]
 

\[ y(x)\to c_2 \left (\frac {7 x^5}{72}-\frac {5 x^4}{54}+\frac {5 x^3}{18}+x\right )+c_1 \left (-\frac {x^5}{9}+\frac {16 x^4}{81}-\frac {4 x^3}{27}+\frac {2 x^2}{3}+1\right ) \]

✓ Sympy. Time used: 0.971 (sec). Leaf size: 48

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=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {16 x^{4}}{81} - \frac {4 x^{3}}{27} + \frac {2 x^{2}}{3} + 1\right ) + C_{1} x \left (- \frac {5 x^{3}}{54} + \frac {5 x^{2}}{18} + 1\right ) + O\left (x^{6}\right ) \]