54.4.3 problem 3

Internal problem ID [8587]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 18. Power series solutions. 18.4 Indicial Equation with Difference of Roots Nonintegral. Exercises page 365
Problem number : 3
Date solved : Wednesday, March 05, 2025 at 06:10:04 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} y^{\prime \prime }+4 x y^{\prime }-\left (4 x^{2}+1\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.019 (sec). Leaf size: 38

Order:=8; 
ode:=4*x^2*diff(diff(y(x),x),x)+4*x*diff(y(x),x)-(4*x^2+1)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_{1} x \left (1+\frac {1}{6} x^{2}+\frac {1}{120} x^{4}+\frac {1}{5040} x^{6}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (1+\frac {1}{2} x^{2}+\frac {1}{24} x^{4}+\frac {1}{720} x^{6}+\operatorname {O}\left (x^{8}\right )\right )}{\sqrt {x}} \]

✓ Mathematica. Time used: 0.019 (sec). Leaf size: 76

ode=4*x^2*D[y[x],{x,2}]+4*x*D[y[x],x]-(4*x^2+1)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_1 \left (\frac {x^{11/2}}{720}+\frac {x^{7/2}}{24}+\frac {x^{3/2}}{2}+\frac {1}{\sqrt {x}}\right )+c_2 \left (\frac {x^{13/2}}{5040}+\frac {x^{9/2}}{120}+\frac {x^{5/2}}{6}+\sqrt {x}\right ) \]

✓ Sympy. Time used: 0.989 (sec). Leaf size: 53

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) - (4*x**2 + 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{2} \sqrt {x} \left (\frac {x^{6}}{5040} + \frac {x^{4}}{120} + \frac {x^{2}}{6} + 1\right ) + \frac {C_{1} \left (\frac {x^{6}}{720} + \frac {x^{4}}{24} + \frac {x^{2}}{2} + 1\right )}{\sqrt {x}} + O\left (x^{8}\right ) \]