20.24.9 problem Problem 9

Internal problem ID [3994]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 11, Series Solutions to Linear Differential Equations. Exercises for 11.2. page 739
Problem number : Problem 9
Date solved : Sunday, March 30, 2025 at 02:14:02 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} \left (x^{2}-3\right ) y^{\prime \prime }-3 x y^{\prime }-5 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)-5*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

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

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 42

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

\[ y(x)\to c_2 \left (\frac {8 x^5}{135}-\frac {4 x^3}{9}+x\right )+c_1 \left (\frac {5 x^4}{24}-\frac {5 x^2}{6}+1\right ) \]

✓ Sympy. Time used: 0.861 (sec). Leaf size: 34

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-3*x*Derivative(y(x), x) + (x**2 - 3)*Derivative(y(x), (x, 2)) - 5*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {5 x^{4}}{24} - \frac {5 x^{2}}{6} + 1\right ) + C_{1} x \left (1 - \frac {4 x^{2}}{9}\right ) + O\left (x^{6}\right ) \]