54.7.5 problem 5

Internal problem ID [8653]
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 : 5
Date solved : Sunday, March 30, 2025 at 01:22:27 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-x \left (6+x \right ) y^{\prime }+10 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.026 (sec). Leaf size: 74

Order:=8; 
ode:=x^2*diff(diff(y(x),x),x)-x*(6+x)*diff(y(x),x)+10*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = x^{2} \left (c_1 \,x^{3} \left (1+\frac {5}{4} x +\frac {3}{4} x^{2}+\frac {7}{24} x^{3}+\frac {1}{12} x^{4}+\frac {3}{160} x^{5}+\frac {1}{288} x^{6}+\frac {11}{20160} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_2 \left (\left (24 x^{3}+30 x^{4}+18 x^{5}+7 x^{6}+2 x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \ln \left (x \right )+\left (12-12 x +18 x^{2}+26 x^{3}+x^{4}-9 x^{5}-6 x^{6}-\frac {9}{4} x^{7}+\operatorname {O}\left (x^{8}\right )\right )\right )\right ) \]

✓ Mathematica. Time used: 0.054 (sec). Leaf size: 118

ode=x^2*D[y[x],{x,2}]-x*(6+x)*D[y[x],x]+10*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_1 \left (\frac {1}{12} x^5 \left (7 x^3+18 x^2+30 x+24\right ) \log (x)-\frac {1}{36} x^2 \left (25 x^6+45 x^5+27 x^4-54 x^3-54 x^2+36 x-36\right )\right )+c_2 \left (\frac {x^{11}}{288}+\frac {3 x^{10}}{160}+\frac {x^9}{12}+\frac {7 x^8}{24}+\frac {3 x^7}{4}+\frac {5 x^6}{4}+x^5\right ) \]

✓ Sympy. Time used: 0.811 (sec). Leaf size: 24

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) - x*(x + 6)*Derivative(y(x), x) + 10*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

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