54.5.4 problem 4

Internal problem ID [8619]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 18. Power series solutions. 18.6. Indicial Equation with Equal Roots. Exercises page 373
Problem number : 4
Date solved : Sunday, March 30, 2025 at 01:21:26 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+3 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: 0.030 (sec). Leaf size: 40

Order:=8; 
ode:=x^2*diff(diff(y(x),x),x)+3*x*diff(y(x),x)+(4*x^2+1)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1-x^{2}+\frac {1}{4} x^{4}-\frac {1}{36} x^{6}+\operatorname {O}\left (x^{8}\right )\right )+\left (x^{2}-\frac {3}{8} x^{4}+\frac {11}{216} x^{6}+\operatorname {O}\left (x^{8}\right )\right ) c_2}{x} \]

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 84

ode=x^2*D[y[x],{x,2}]+3*x*D[y[x],x]+(1+4*x^2)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to \frac {c_1 \left (-\frac {x^6}{36}+\frac {x^4}{4}-x^2+1\right )}{x}+c_2 \left (\frac {\frac {11 x^6}{216}-\frac {3 x^4}{8}+x^2}{x}+\frac {\left (-\frac {x^6}{36}+\frac {x^4}{4}-x^2+1\right ) \log (x)}{x}\right ) \]

✓ Sympy. Time used: 0.856 (sec). Leaf size: 29

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + 3*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)
 

\[ y{\left (x \right )} = \frac {C_{1} \left (\frac {x^{8}}{576} - \frac {x^{6}}{36} + \frac {x^{4}}{4} - x^{2} + 1\right )}{x} + O\left (x^{8}\right ) \]