54.3.21 problem 21

Internal problem ID [8577]
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 : 21
Date solved : Wednesday, March 05, 2025 at 06:09:54 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} \left (x^{2}+4\right ) y^{\prime \prime }+x y^{\prime }-9 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 39

Order:=8; 
ode:=(x^2+4)*diff(diff(y(x),x),x)+x*diff(y(x),x)-9*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1+\frac {9}{8} x^{2}+\frac {15}{128} x^{4}-\frac {7}{1024} x^{6}\right ) y \left (0\right )+\left (\frac {1}{3} x^{3}+x \right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 42

ode=(x^2+4)*D[y[x],{x,2}]+x*D[y[x],x]-9*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_2 \left (\frac {x^3}{3}+x\right )+c_1 \left (-\frac {7 x^6}{1024}+\frac {15 x^4}{128}+\frac {9 x^2}{8}+1\right ) \]

✓ Sympy. Time used: 0.781 (sec). Leaf size: 39

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*Derivative(y(x), x) + (x**2 + 4)*Derivative(y(x), (x, 2)) - 9*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{2} \left (- \frac {7 x^{6}}{1024} + \frac {15 x^{4}}{128} + \frac {9 x^{2}}{8} + 1\right ) + C_{1} x \left (\frac {x^{2}}{3} + 1\right ) + O\left (x^{8}\right ) \]