10.13.9 problem 11

Internal problem ID [1369]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 5.2, Series Solutions Near an Ordinary Point, Part I. page 263
Problem number : 11
Date solved : Saturday, March 29, 2025 at 10:53:31 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 39

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

\[ y = \left (1+\frac {1}{6} x^{2}+\frac {1}{24} x^{4}\right ) y \left (0\right )+\left (x +\frac {2}{9} x^{3}+\frac {8}{135} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 70

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

\[ y(x)\to c_1 \left (\frac {13 x^5}{1080}+\frac {x^4}{36}+\frac {x^3}{18}+\frac {x^2}{6}+1\right )+c_2 \left (\frac {49 x^5}{1080}+\frac {7 x^4}{72}+\frac {2 x^3}{9}+\frac {x^2}{2}+x\right ) \]

✓ Sympy. Time used: 0.836 (sec). Leaf size: 31

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

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{4}}{24} + \frac {x^{2}}{6} + 1\right ) + C_{1} x \left (\frac {2 x^{2}}{9} + 1\right ) + O\left (x^{6}\right ) \]