48.6.1 problem 1

Internal problem ID [9919]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 18. Power series solutions. 18.8 Indicial Equation with Difference of Roots a Positive Integer: Nonlogarithmic Case. Exercises page 380
Problem number : 1
Date solved : Tuesday, September 30, 2025 at 06:44:25 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.023 (sec). Leaf size: 53

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

\[ y = c_1 \,x^{4} \left (1-\frac {1}{2} x +\frac {1}{5} x^{2}-\frac {1}{15} x^{3}+\frac {2}{105} x^{4}-\frac {1}{210} x^{5}+\frac {1}{945} x^{6}-\frac {1}{4725} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_2 x \left (12-24 x +24 x^{2}-16 x^{3}+8 x^{4}-\frac {16}{5} x^{5}+\frac {16}{15} x^{6}-\frac {32}{105} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Mathematica. Time used: 0.064 (sec). Leaf size: 96

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

\[ y(x)\to c_1 \left (\frac {4 x^7}{45}-\frac {4 x^6}{15}+\frac {2 x^5}{3}-\frac {4 x^4}{3}+2 x^3-2 x^2+x\right )+c_2 \left (\frac {x^{10}}{945}-\frac {x^9}{210}+\frac {2 x^8}{105}-\frac {x^7}{15}+\frac {x^6}{5}-\frac {x^5}{2}+x^4\right ) \]

✓ Sympy. Time used: 0.327 (sec). Leaf size: 39

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + 2*x*(x - 2)*Derivative(y(x), x) + (4 - 6*x)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{2} x^{4} \left (- \frac {x^{3}}{15} + \frac {x^{2}}{5} - \frac {x}{2} + 1\right ) + C_{1} x \left (2 x^{2} - 2 x + 1\right ) + O\left (x^{8}\right ) \]