54.3.20 problem 20

Internal problem ID [8576]
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 : 20
Date solved : Wednesday, March 05, 2025 at 06:09:53 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 y^{\prime \prime }+9 x y^{\prime }-36 y&=0 \end{align*}

Using series method with expansion around

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

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

Order:=8; 
ode:=2*diff(diff(y(x),x),x)+9*x*diff(y(x),x)-36*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (\frac {27}{4} x^{4}+9 x^{2}+1\right ) y \left (0\right )+\left (x +\frac {9}{4} x^{3}+\frac {81}{160} x^{5}-\frac {243}{4480} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 47

ode=2*D[y[x],{x,2}]+9*x*D[y[x],x]-36*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_1 \left (\frac {27 x^4}{4}+9 x^2+1\right )+c_2 \left (-\frac {243 x^7}{4480}+\frac {81 x^5}{160}+\frac {9 x^3}{4}+x\right ) \]

✓ Sympy. Time used: 0.789 (sec). Leaf size: 39

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(9*x*Derivative(y(x), x) - 36*y(x) + 2*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {27 x^{4}}{4} + 9 x^{2} + 1\right ) + C_{1} x \left (\frac {81 x^{4}}{160} + \frac {9 x^{2}}{4} + 1\right ) + O\left (x^{8}\right ) \]