54.5.7 problem 7

Internal problem ID [8622]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 18. Power series solutions. 18.6. Indicial Equation with Equal Roots. Exercises page 373
Problem number : 7
Date solved : Sunday, March 30, 2025 at 01:21:31 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.024 (sec). Leaf size: 40

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

\[ y = x \left (\left (-x +\frac {1}{4} x^{2}-\frac {1}{18} x^{3}+\frac {1}{96} x^{4}-\frac {1}{600} x^{5}+\frac {1}{4320} x^{6}-\frac {1}{35280} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_2 +\left (1+\operatorname {O}\left (x^{8}\right )\right ) \left (c_2 \ln \left (x \right )+c_1 \right )\right ) \]

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 64

ode=x^2*D[y[x],{x,2}]+x*(x-1)*D[y[x],x]+(1-x)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_2 \left (x \left (-\frac {x^7}{35280}+\frac {x^6}{4320}-\frac {x^5}{600}+\frac {x^4}{96}-\frac {x^3}{18}+\frac {x^2}{4}-x\right )+x \log (x)\right )+c_1 x \]

✓ Sympy. Time used: 0.906 (sec). Leaf size: 8

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

\[ y{\left (x \right )} = C_{1} x + O\left (x^{8}\right ) \]