54.4.18 problem 19

Internal problem ID [8602]
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 : 19
Date solved : Wednesday, March 05, 2025 at 06:10:24 AM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} 2 x^{2} y^{\prime \prime }+x y^{\prime }-y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.012 (sec). Leaf size: 20

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

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

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 16

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

\[ y(x)\to c_1 x+\frac {c_2}{\sqrt {x}} \]

✓ Sympy. Time used: 0.744 (sec). Leaf size: 15

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

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