54.6.13 problem 13

Internal problem ID [8645]
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 : 13
Date solved : Wednesday, March 05, 2025 at 06:11:22 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x \left (x +3\right ) y^{\prime \prime }-9 y^{\prime }-6 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.021 (sec). Leaf size: 52

Order:=8; 
ode:=x*(x+3)*diff(diff(y(x),x),x)-9*diff(y(x),x)-6*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_{1} x^{4} \left (1-\frac {2}{5} x +\frac {7}{45} x^{2}-\frac {8}{135} x^{3}+\frac {1}{45} x^{4}-\frac {2}{243} x^{5}+\frac {11}{3645} x^{6}-\frac {4}{3645} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (-144+96 x -48 x^{2}+\frac {64}{3} x^{3}-\frac {80}{9} x^{4}+\frac {32}{9} x^{5}-\frac {112}{81} x^{6}+\frac {128}{243} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Mathematica. Time used: 0.386 (sec). Leaf size: 98

ode=x*(x+3)*D[y[x],{x,2}]-9*D[y[x],x]-6*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_1 \left (\frac {7 x^6}{729}-\frac {2 x^5}{81}+\frac {5 x^4}{81}-\frac {4 x^3}{27}+\frac {x^2}{3}-\frac {2 x}{3}+1\right )+c_2 \left (\frac {11 x^{10}}{3645}-\frac {2 x^9}{243}+\frac {x^8}{45}-\frac {8 x^7}{135}+\frac {7 x^6}{45}-\frac {2 x^5}{5}+x^4\right ) \]

✓ Sympy. Time used: 0.954 (sec). Leaf size: 31

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*(x + 3)*Derivative(y(x), (x, 2)) - 6*y(x) - 9*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{1} x^{4} \left (\frac {6 x^{3}}{35} + \frac {3 x^{2}}{5} + \frac {6 x}{5} + 1\right ) + O\left (x^{8}\right ) \]