48.9.24 problem 25

Internal problem ID [9980]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 18. Power series solutions. Miscellaneous Exercises. page 394
Problem number : 25
Date solved : Tuesday, September 30, 2025 at 06:45:32 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x \left (1-2 x \right ) y^{\prime \prime }-2 \left (2+x \right ) y^{\prime }+18 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.037 (sec). Leaf size: 50

Order:=8; 
ode:=x*(1-2*x)*diff(diff(y(x),x),x)-2*(x+2)*diff(y(x),x)+18*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{5} \left (1+\frac {16}{3} x +\frac {144}{7} x^{2}+\frac {480}{7} x^{3}+\frac {4400}{21} x^{4}+\frac {4224}{7} x^{5}+1664 x^{6}+\frac {13312}{3} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_2 \left (2880+12960 x +34560 x^{2}+57600 x^{3}-483840 x^{5}-2580480 x^{6}-9953280 x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Mathematica. Time used: 0.074 (sec). Leaf size: 81

ode=x*(1-2*x)*D[y[x],{x,2}]-2*(2+x)*D[y[x],x]+18*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_1 \left (-896 x^6-168 x^5+20 x^3+12 x^2+\frac {9 x}{2}+1\right )+c_2 \left (1664 x^{11}+\frac {4224 x^{10}}{7}+\frac {4400 x^9}{21}+\frac {480 x^8}{7}+\frac {144 x^7}{7}+\frac {16 x^6}{3}+x^5\right ) \]

✓ Sympy. Time used: 0.378 (sec). Leaf size: 22

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*(1 - 2*x)*Derivative(y(x), (x, 2)) - (2*x + 4)*Derivative(y(x), x) + 18*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{1} x^{5} \left (\frac {27 x^{2}}{7} + 3 x + 1\right ) + O\left (x^{8}\right ) \]