54.4.15 problem 15

Internal problem ID [8599]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 18. Power series solutions. 18.4 Indicial Equation with Difference of Roots Nonintegral. Exercises page 365
Problem number : 15
Date solved : Sunday, March 30, 2025 at 01:20:53 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 x^{2} y^{\prime \prime }-3 x \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: 0.023 (sec). Leaf size: 55

Order:=8; 
ode:=2*x^2*diff(diff(y(x),x),x)-3*x*(1-x)*diff(y(x),x)+2*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \sqrt {x}\, \left (1+\frac {3}{2} x -\frac {27}{8} x^{2}+\frac {45}{16} x^{3}-\frac {189}{128} x^{4}+\frac {729}{1280} x^{5}-\frac {891}{5120} x^{6}+\frac {3159}{71680} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_2 \,x^{2} \left (1-\frac {6}{5} x +\frac {27}{35} x^{2}-\frac {12}{35} x^{3}+\frac {9}{77} x^{4}-\frac {162}{5005} x^{5}+\frac {27}{3575} x^{6}-\frac {648}{425425} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 116

ode=2*x^2*D[y[x],{x,2}]-3*x*(1-x)*D[y[x],x]+2*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_1 \left (-\frac {648 x^7}{425425}+\frac {27 x^6}{3575}-\frac {162 x^5}{5005}+\frac {9 x^4}{77}-\frac {12 x^3}{35}+\frac {27 x^2}{35}-\frac {6 x}{5}+1\right ) x^2+c_2 \left (\frac {3159 x^7}{71680}-\frac {891 x^6}{5120}+\frac {729 x^5}{1280}-\frac {189 x^4}{128}+\frac {45 x^3}{16}-\frac {27 x^2}{8}+\frac {3 x}{2}+1\right ) \sqrt {x} \]

✓ Sympy. Time used: 1.009 (sec). Leaf size: 92

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x**2*Derivative(y(x), (x, 2)) - 3*x*(1 - 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)
 

\[ y{\left (x \right )} = C_{2} x^{2} \left (- \frac {162 x^{5}}{5005} + \frac {9 x^{4}}{77} - \frac {12 x^{3}}{35} + \frac {27 x^{2}}{35} - \frac {6 x}{5} + 1\right ) + C_{1} \sqrt {x} \left (- \frac {891 x^{6}}{5120} + \frac {729 x^{5}}{1280} - \frac {189 x^{4}}{128} + \frac {45 x^{3}}{16} - \frac {27 x^{2}}{8} + \frac {3 x}{2} + 1\right ) + O\left (x^{8}\right ) \]