12.14.29 problem 31

Internal problem ID [1970]
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 : 31
Date solved : Tuesday, March 04, 2025 at 01:47:13 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 46

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

\[ y = \frac {c_2 \,x^{{5}/{2}} \left (1-\frac {3}{4} x +\frac {5}{96} x^{2}+\frac {5}{4224} x^{3}+\frac {5}{292864} x^{4}-\frac {1}{3514368} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_1 \left (1+8 x +60 x^{2}-160 x^{3}+40 x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{2}} \]

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 73

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

\[ y(x)\to \frac {c_2 \left (40 x^4-160 x^3+60 x^2+8 x+1\right )}{x^2}+c_1 \sqrt {x} \left (-\frac {x^5}{3514368}+\frac {5 x^4}{292864}+\frac {5 x^3}{4224}+\frac {5 x^2}{96}-\frac {3 x}{4}+1\right ) \]

✓ Sympy. Time used: 1.093 (sec). Leaf size: 17

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

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