6.15.63 problem 64

Internal problem ID [2061]
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 : 64
Date solved : Tuesday, September 30, 2025 at 05:23:09 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.021 (sec). Leaf size: 42

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

\[ y = \left (6 x^{5}+5 x^{4}+4 x^{3}+3 x^{2}+2 x +1\right ) x \left (\ln \left (x \right ) c_2 +c_1 \right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 64

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

\[ y(x)\to c_1 x \left (6 x^5+5 x^4+4 x^3+3 x^2+2 x+1\right )+c_2 x \left (6 x^5+5 x^4+4 x^3+3 x^2+2 x+1\right ) \log (x) \]

✓ Sympy. Time used: 0.388 (sec). Leaf size: 8

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

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