12.14.39 problem 41

Internal problem ID [1980]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.5 THE METHOD OF FROBENIUS I. Exercises 7.5. Page 358
Problem number : 41
Date solved : Tuesday, March 04, 2025 at 01:47:25 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 x^{2} \left (2 x^{2}+1\right ) y^{\prime \prime }+5 x \left (6 x^{2}+1\right ) y^{\prime }-\left (-40 x^{2}+2\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 36

Order:=6; 
ode:=2*x^2*(2*x^2+1)*diff(diff(y(x),x),x)+5*x*(6*x^2+1)*diff(y(x),x)-(-40*x^2+2)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_2 \,x^{{5}/{2}} \left (1-3 x^{2}+\frac {15}{2} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_1 \left (1+2 x^{2}-\frac {20}{3} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{2}} \]

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 46

ode=2*x^2*(1+2*x^2)*D[y[x],{x,2}]+5*x*(1+6*x^2)*D[y[x],x]-(2-40*x^2)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \sqrt {x} \left (\frac {15 x^4}{2}-3 x^2+1\right )+\frac {c_2 \left (-\frac {20 x^4}{3}+2 x^2+1\right )}{x^2} \]

✓ Sympy. Time used: 1.226 (sec). Leaf size: 17

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x**2*(2*x**2 + 1)*Derivative(y(x), (x, 2)) + 5*x*(6*x**2 + 1)*Derivative(y(x), x) - (2 - 40*x**2)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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