54.3.7 problem 7

Internal problem ID [8563]
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 : 7
Date solved : Wednesday, March 05, 2025 at 06:09:40 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{2}+1\right ) y^{\prime \prime }+10 x y^{\prime }+20 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 49

Order:=8; 
ode:=(x^2+1)*diff(diff(y(x),x),x)+10*x*diff(y(x),x)+20*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (-84 x^{6}+35 x^{4}-10 x^{2}+1\right ) y \left (0\right )+\left (-30 x^{7}+14 x^{5}-5 x^{3}+x \right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 44

ode=(1+x^2)*D[y[x],{x,2}]+10*x*D[y[x],x]+20*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_2 \left (-30 x^7+14 x^5-5 x^3+x\right )+c_1 \left (-84 x^6+35 x^4-10 x^2+1\right ) \]

✓ Sympy. Time used: 0.770 (sec). Leaf size: 39

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(10*x*Derivative(y(x), x) + (x**2 + 1)*Derivative(y(x), (x, 2)) + 20*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{2} \left (- 84 x^{6} + 35 x^{4} - 10 x^{2} + 1\right ) + C_{1} x \left (14 x^{4} - 5 x^{2} + 1\right ) + O\left (x^{8}\right ) \]