45.3.9 problem 9

Internal problem ID [7267]
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. 6.3.1 page 250
Problem number : 9
Date solved : Wednesday, March 05, 2025 at 04:22:16 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x y^{\prime }+\left (25 x^{2}-\frac {4}{9}\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 36

Order:=6; 
ode:=x^2*diff(diff(y(x),x),x)+x*diff(y(x),x)+(25*x^2-4/9)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_{2} x^{{4}/{3}} \left (1-\frac {15}{4} x^{2}+\frac {1125}{256} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{1} \left (1-\frac {75}{4} x^{2}+\frac {5625}{128} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{{2}/{3}}} \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 52

ode=x^2*D[y[x],{x,2}]+x*D[y[x],x]+(25*x^2-4/9)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 x^{2/3} \left (\frac {1125 x^4}{256}-\frac {15 x^2}{4}+1\right )+\frac {c_2 \left (\frac {5625 x^4}{128}-\frac {75 x^2}{4}+1\right )}{x^{2/3}} \]

✓ Sympy. Time used: 1.042 (sec). Leaf size: 49

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + x*Derivative(y(x), x) + (25*x**2 - 4/9)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} x^{\frac {2}{3}} \left (\frac {1125 x^{4}}{256} - \frac {15 x^{2}}{4} + 1\right ) + \frac {C_{1} \left (\frac {5625 x^{4}}{128} - \frac {75 x^{2}}{4} + 1\right )}{x^{\frac {2}{3}}} + O\left (x^{6}\right ) \]