54.4.9 problem 9

Internal problem ID [8593]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 18. Power series solutions. 18.4 Indicial Equation with Difference of Roots Nonintegral. Exercises page 365
Problem number : 9
Date solved : Sunday, March 30, 2025 at 01:20:42 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 x \left (x +3\right ) y^{\prime \prime }-3 \left (x +1\right ) y^{\prime }+2 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.036 (sec). Leaf size: 42

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

\[ y = c_1 \,x^{{3}/{2}} \left (1+\frac {1}{15} x -\frac {1}{315} x^{2}+\frac {1}{2835} x^{3}-\frac {1}{18711} x^{4}+\frac {1}{104247} x^{5}-\frac {1}{521235} x^{6}+\frac {1}{2416635} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_2 \left (1+\frac {2}{3} x +\frac {1}{9} x^{2}+\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 78

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

\[ y(x)\to c_2 \left (\frac {x^2}{9}+\frac {2 x}{3}+1\right )+c_1 \left (\frac {x^7}{2416635}-\frac {x^6}{521235}+\frac {x^5}{104247}-\frac {x^4}{18711}+\frac {x^3}{2835}-\frac {x^2}{315}+\frac {x}{15}+1\right ) x^{3/2} \]

✓ Sympy. Time used: 1.030 (sec). Leaf size: 14

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

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