45.2.20 problem 20

Internal problem ID [7243]
Book : A FIRST COURSE IN DIFFERENTIAL EQUATIONS with Modeling Applications. Dennis G. Zill. 9th edition. Brooks/Cole. CA, USA.
Section : Chapter 6. SERIES SOLUTIONS OF LINEAR EQUATIONS. Exercises. 6.2 page 239
Problem number : 20
Date solved : Wednesday, March 05, 2025 at 04:21:46 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 47

Order:=6; 
ode:=x^2*diff(diff(y(x),x),x)-(x-2/9)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_{1} x^{{1}/{3}} \left (1+\frac {3}{2} x +\frac {9}{20} x^{2}+\frac {9}{160} x^{3}+\frac {27}{7040} x^{4}+\frac {81}{492800} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x^{{2}/{3}} \left (1+\frac {3}{4} x +\frac {9}{56} x^{2}+\frac {9}{560} x^{3}+\frac {27}{29120} x^{4}+\frac {81}{2329600} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 90

ode=x^2*D[y[x],{x,2}]-(x-2/9)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \sqrt [3]{x} \left (\frac {81 x^5}{492800}+\frac {27 x^4}{7040}+\frac {9 x^3}{160}+\frac {9 x^2}{20}+\frac {3 x}{2}+1\right )+c_1 x^{2/3} \left (\frac {81 x^5}{2329600}+\frac {27 x^4}{29120}+\frac {9 x^3}{560}+\frac {9 x^2}{56}+\frac {3 x}{4}+1\right ) \]

✓ Sympy. Time used: 1.047 (sec). Leaf size: 73

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) - (x - 2/9)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} x^{\frac {2}{3}} \left (\frac {27 x^{4}}{29120} + \frac {9 x^{3}}{560} + \frac {9 x^{2}}{56} + \frac {3 x}{4} + 1\right ) + C_{1} \sqrt [3]{x} \left (\frac {27 x^{4}}{7040} + \frac {9 x^{3}}{160} + \frac {9 x^{2}}{20} + \frac {3 x}{2} + 1\right ) + O\left (x^{6}\right ) \]