54.7.7 problem 7

Internal problem ID [8655]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 18. Power series solutions. 18.9 Indicial Equation with Difference of Roots a Positive Integer: Logarithmic Case. Exercises page 384
Problem number : 7
Date solved : Sunday, March 30, 2025 at 01:22:32 PM
CAS classification : [_Jacobi]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.032 (sec). Leaf size: 52

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

\[ y = \frac {c_1 \left (1-x +\operatorname {O}\left (x^{8}\right )\right ) x +\left (\left (-2\right ) x +2 x^{2}+\operatorname {O}\left (x^{8}\right )\right ) \ln \left (x \right ) c_2 +\left (1-4 x^{2}+x^{3}+\frac {1}{3} x^{4}+\frac {1}{6} x^{5}+\frac {1}{10} x^{6}+\frac {1}{15} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_2}{x} \]

✓ Mathematica. Time used: 0.382 (sec). Leaf size: 60

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

\[ y(x)\to c_1 \left (\frac {3 x^6+5 x^5+10 x^4+30 x^3-150 x^2+30 x+30}{30 x}+2 (x-1) \log (x)\right )+c_2 (1-x) \]

✓ Sympy. Time used: 0.931 (sec). Leaf size: 41

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

\[ y{\left (x \right )} = C_{1} \left (\frac {x^{7}}{1587600} + \frac {x^{6}}{56700} + \frac {x^{5}}{2700} + \frac {x^{4}}{180} + \frac {x^{3}}{18} + \frac {x^{2}}{3} + x + 1\right ) + O\left (x^{8}\right ) \]