6.15.19 problem 15

Internal problem ID [2017]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.6 THE METHOD OF FROBENIUS II. Exercises 7.6. Page 374
Problem number : 15
Date solved : Tuesday, September 30, 2025 at 05:22:35 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (1-2 x \right ) y^{\prime \prime }-x \left (5-4 x \right ) y^{\prime }+\left (9-4 x \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.029 (sec). Leaf size: 48

Order:=6; 
ode:=x^2*(1-2*x)*diff(diff(y(x),x),x)-x*(5-4*x)*diff(y(x),x)+(9-4*x)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1+4 x +12 x^{2}+32 x^{3}+80 x^{4}+192 x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (\left (-2\right ) x -8 x^{2}-24 x^{3}-64 x^{4}-160 x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \right ) x^{3} \]

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 98

ode=x^2*(1-2*x)*D[y[x],{x,2}]-x*(5-4*x)*D[y[x],x]+(9-4*x)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (192 x^5+80 x^4+32 x^3+12 x^2+4 x+1\right ) x^3+c_2 \left (\left (-160 x^5-64 x^4-24 x^3-8 x^2-2 x\right ) x^3+\left (192 x^5+80 x^4+32 x^3+12 x^2+4 x+1\right ) x^3 \log (x)\right ) \]

✓ Sympy. Time used: 0.424 (sec). Leaf size: 10

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(1 - 2*x)*Derivative(y(x), (x, 2)) - x*(5 - 4*x)*Derivative(y(x), x) + (9 - 4*x)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{1} x^{3} + O\left (x^{6}\right ) \]