6.15.18 problem 14

Internal problem ID [2016]
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 : 14
Date solved : Tuesday, September 30, 2025 at 05:22:35 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (1+x \right ) y^{\prime \prime }-x \left (3-x \right ) y^{\prime }+4 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.026 (sec). Leaf size: 48

Order:=6; 
ode:=x^2*(1+x)*diff(diff(y(x),x),x)-x*(3-x)*diff(y(x),x)+4*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1-4 x +9 x^{2}-16 x^{3}+25 x^{4}-36 x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (4 x -12 x^{2}+24 x^{3}-40 x^{4}+60 x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \right ) x^{2} \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 98

ode=x^2*(1+x)*D[y[x],{x,2}]-x*(3-x)*D[y[x],x]+4*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (-36 x^5+25 x^4-16 x^3+9 x^2-4 x+1\right ) x^2+c_2 \left (\left (60 x^5-40 x^4+24 x^3-12 x^2+4 x\right ) x^2+\left (-36 x^5+25 x^4-16 x^3+9 x^2-4 x+1\right ) x^2 \log (x)\right ) \]

✓ Sympy. Time used: 0.357 (sec). Leaf size: 10

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(x + 1)*Derivative(y(x), (x, 2)) - x*(3 - x)*Derivative(y(x), x) + 4*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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