54.5.6 problem 6

Internal problem ID [8621]
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 : 6
Date solved : Sunday, March 30, 2025 at 01:21:29 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-x \left (1+3 x \right ) y^{\prime }+\left (1-6 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: 54

Order:=8; 
ode:=x^2*diff(diff(y(x),x),x)-x*(3*x+1)*diff(y(x),x)+(1-6*x)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1+9 x +27 x^{2}+45 x^{3}+\frac {405}{8} x^{4}+\frac {1701}{40} x^{5}+\frac {567}{20} x^{6}+\frac {2187}{140} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (\left (-15\right ) x -\frac {261}{4} x^{2}-\frac {519}{4} x^{3}-\frac {5211}{32} x^{4}-\frac {118179}{800} x^{5}-\frac {83511}{800} x^{6}-\frac {2361717}{39200} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_2 \right ) x \]

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 150

ode=x^2*D[y[x],{x,2}]-x*(1+3*x)*D[y[x],x]+(1-6*x)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_1 x \left (\frac {2187 x^7}{140}+\frac {567 x^6}{20}+\frac {1701 x^5}{40}+\frac {405 x^4}{8}+45 x^3+27 x^2+9 x+1\right )+c_2 \left (x \left (-\frac {2361717 x^7}{39200}-\frac {83511 x^6}{800}-\frac {118179 x^5}{800}-\frac {5211 x^4}{32}-\frac {519 x^3}{4}-\frac {261 x^2}{4}-15 x\right )+x \left (\frac {2187 x^7}{140}+\frac {567 x^6}{20}+\frac {1701 x^5}{40}+\frac {405 x^4}{8}+45 x^3+27 x^2+9 x+1\right ) \log (x)\right ) \]

✓ Sympy. Time used: 0.996 (sec). Leaf size: 44

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) - x*(3*x + 1)*Derivative(y(x), x) + (1 - 6*x)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{1} x \left (\frac {567 x^{6}}{20} + \frac {1701 x^{5}}{40} + \frac {405 x^{4}}{8} + 45 x^{3} + 27 x^{2} + 9 x + 1\right ) + O\left (x^{8}\right ) \]