45.3.24 problem 26

Internal problem ID [7282]
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 : 26
Date solved : Sunday, March 30, 2025 at 11:53:20 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} y^{\prime \prime }-4 x y^{\prime }+\left (16 x^{4}+3\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.024 (sec). Leaf size: 30

Order:=6; 
ode:=4*x^2*diff(diff(y(x),x),x)-4*x*diff(y(x),x)+(16*x^4+3)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \sqrt {x}\, \left (x \left (1-\frac {1}{5} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_1 +\left (1-\frac {1}{3} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \right ) \]

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 40

ode=4*x^2*D[y[x],{x,2}]-4*x*D[y[x],x]+(16*x^4+3)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\sqrt {x}-\frac {x^{9/2}}{3}\right )+c_2 \left (x^{3/2}-\frac {x^{11/2}}{5}\right ) \]

✓ Sympy. Time used: 0.959 (sec). Leaf size: 26

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*x**2*Derivative(y(x), (x, 2)) - 4*x*Derivative(y(x), x) + (16*x**4 + 3)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \sqrt {x} \left (1 - \frac {x^{4}}{3}\right ) + C_{1} x^{\frac {3}{2}} + O\left (x^{6}\right ) \]