54.9.11 problem 11

Internal problem ID [8681]
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 : 11
Date solved : Sunday, March 30, 2025 at 01:23:24 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} y^{\prime \prime }-2 x \left (2+x \right ) y^{\prime }+\left (3+x \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.029 (sec). Leaf size: 54

Order:=8; 
ode:=4*x^2*diff(diff(y(x),x),x)-2*x*(x+2)*diff(y(x),x)+(x+3)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \sqrt {x}\, \left (\left (1+\frac {1}{4} x +\frac {1}{24} x^{2}+\frac {1}{192} x^{3}+\frac {1}{1920} x^{4}+\frac {1}{23040} x^{5}+\frac {1}{322560} x^{6}+\frac {1}{5160960} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) x c_1 +\left (1+\frac {1}{2} x +\frac {1}{8} x^{2}+\frac {1}{48} x^{3}+\frac {1}{384} x^{4}+\frac {1}{3840} x^{5}+\frac {1}{46080} x^{6}+\frac {1}{645120} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_2 \right ) \]

✓ Mathematica. Time used: 0.103 (sec). Leaf size: 130

ode=4*x^2*D[y[x],{x,2}]-2*x*(2+x)*D[y[x],x]+(3+x)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_1 \left (\frac {x^{13/2}}{46080}+\frac {x^{11/2}}{3840}+\frac {x^{9/2}}{384}+\frac {x^{7/2}}{48}+\frac {x^{5/2}}{8}+\frac {x^{3/2}}{2}+\sqrt {x}\right )+c_2 \left (\frac {x^{15/2}}{322560}+\frac {x^{13/2}}{23040}+\frac {x^{11/2}}{1920}+\frac {x^{9/2}}{192}+\frac {x^{7/2}}{24}+\frac {x^{5/2}}{4}+x^{3/2}\right ) \]

✓ Sympy. Time used: 0.955 (sec). Leaf size: 44

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*x**2*Derivative(y(x), (x, 2)) - 2*x*(x + 2)*Derivative(y(x), x) + (x + 3)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{2} x^{\frac {3}{2}} \left (\frac {x^{5}}{23040} + \frac {x^{4}}{1920} + \frac {x^{3}}{192} + \frac {x^{2}}{24} + \frac {x}{4} + 1\right ) + C_{1} \sqrt {x} + O\left (x^{8}\right ) \]