12.14.49 problem 51

Internal problem ID [1990]
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 : 51
Date solved : Tuesday, March 04, 2025 at 01:47:37 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 8 x^{2} \left (2 x^{2}+1\right ) y^{\prime \prime }+2 x \left (34 x^{2}+5\right ) y^{\prime }-\left (-30 x^{2}+1\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 36

Order:=6; 
ode:=8*x^2*(2*x^2+1)*diff(diff(y(x),x),x)+2*x*(34*x^2+5)*diff(y(x),x)-(-30*x^2+1)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_2 \,x^{{3}/{4}} \left (1-x^{2}+\frac {3}{2} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_1 \left (1-\frac {2}{5} x^{2}+\frac {36}{65} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}} \]

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 50

ode=8*x^2*(1+2*x^2)*D[y[x],{x,2}]+2*x*(5+34*x^2)*D[y[x],x]-(1-30*x^2)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \sqrt [4]{x} \left (\frac {3 x^4}{2}-x^2+1\right )+\frac {c_2 \left (\frac {36 x^4}{65}-\frac {2 x^2}{5}+1\right )}{\sqrt {x}} \]

✓ Sympy. Time used: 1.191 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(8*x**2*(2*x**2 + 1)*Derivative(y(x), (x, 2)) + 2*x*(34*x**2 + 5)*Derivative(y(x), x) - (1 - 30*x**2)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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