54.9.20 problem 21

Internal problem ID [8690]
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 : 21
Date solved : Wednesday, March 05, 2025 at 06:12:30 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.016 (sec). Leaf size: 56

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

\[ y = \frac {c_{1} x^{2} \left (1-\frac {1}{6} x +\frac {1}{48} x^{2}-\frac {1}{480} x^{3}+\frac {1}{5760} x^{4}-\frac {1}{80640} x^{5}+\frac {1}{1290240} x^{6}-\frac {1}{23224320} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (-2+x -\frac {1}{4} x^{2}+\frac {1}{24} x^{3}-\frac {1}{192} x^{4}+\frac {1}{1920} x^{5}-\frac {1}{23040} x^{6}+\frac {1}{322560} x^{7}+\operatorname {O}\left (x^{8}\right )\right )}{\sqrt {x}} \]

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

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

\[ y(x)\to c_1 \left (\frac {x^{11/2}}{46080}-\frac {x^{9/2}}{3840}+\frac {x^{7/2}}{384}-\frac {x^{5/2}}{48}+\frac {x^{3/2}}{8}-\frac {\sqrt {x}}{2}+\frac {1}{\sqrt {x}}\right )+c_2 \left (\frac {x^{15/2}}{1290240}-\frac {x^{13/2}}{80640}+\frac {x^{11/2}}{5760}-\frac {x^{9/2}}{480}+\frac {x^{7/2}}{48}-\frac {x^{5/2}}{6}+x^{3/2}\right ) \]

✓ Sympy. Time used: 0.974 (sec). Leaf size: 49

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

\[ y{\left (x \right )} = C_{2} x^{\frac {3}{2}} \left (- \frac {x^{5}}{80640} + \frac {x^{4}}{5760} - \frac {x^{3}}{480} + \frac {x^{2}}{48} - \frac {x}{6} + 1\right ) + \frac {C_{1} \left (1 - \frac {x}{2}\right )}{\sqrt {x}} + O\left (x^{8}\right ) \]