54.3.6 problem 6

Internal problem ID [8562]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 17. Power series solutions. 17.5. Solutions Near an Ordinary Point. Exercises page 355
Problem number : 6
Date solved : Wednesday, March 05, 2025 at 06:09:39 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{2}+1\right ) y^{\prime \prime }-4 x y^{\prime }+6 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 25

Order:=8; 
ode:=(x^2+1)*diff(diff(y(x),x),x)-4*x*diff(y(x),x)+6*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = y \left (0\right )+y^{\prime }\left (0\right ) x -3 y \left (0\right ) x^{2}-\frac {y^{\prime }\left (0\right ) x^{3}}{3} \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 26

ode=(1+x^2)*D[y[x],{x,2}]-4*x*D[y[x],x]+6*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_2 \left (x-\frac {x^3}{3}\right )+c_1 \left (1-3 x^2\right ) \]

✓ Sympy. Time used: 0.810 (sec). Leaf size: 24

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-4*x*Derivative(y(x), x) + (x**2 + 1)*Derivative(y(x), (x, 2)) + 6*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{2} \left (1 - 3 x^{2}\right ) + C_{1} x \left (1 - \frac {x^{2}}{3}\right ) + O\left (x^{8}\right ) \]