12.14.54 problem 65

Internal problem ID [1995]
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 : 65
Date solved : Tuesday, March 04, 2025 at 01:47:44 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 8 x^{2} \left (-x^{2}+2\right ) y^{\prime \prime }+2 x \left (-21 x^{2}+10\right ) y^{\prime }-\left (35 x^{2}+2\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 32

Order:=6; 
ode:=8*x^2*(-x^2+2)*diff(diff(y(x),x),x)+2*x*(-21*x^2+10)*diff(y(x),x)-(35*x^2+2)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

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

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 52

ode=8*x^2*(2-x^2)*D[y[x],{x,2}]+2*x*(10-21*x^2)*D[y[x],x]-(2+35*x^2)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

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

✓ Sympy. Time used: 1.088 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(8*x**2*(2 - x**2)*Derivative(y(x), (x, 2)) + 2*x*(10 - 21*x**2)*Derivative(y(x), x) - (35*x**2 + 2)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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