54.4.17 problem 17

Internal problem ID [8601]
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 : 17
Date solved : Sunday, March 30, 2025 at 01:20:57 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.024 (sec). Leaf size: 36

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

\[ y = c_1 \,x^{{3}/{2}} \left (1+\frac {2}{7} x^{2}+\frac {4}{77} x^{4}+\frac {8}{1155} x^{6}+\operatorname {O}\left (x^{8}\right )\right )+c_2 \left (1+\frac {1}{2} x^{2}+\frac {1}{8} x^{4}+\frac {1}{48} x^{6}+\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 61

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

\[ y(x)\to c_2 \left (\frac {x^6}{48}+\frac {x^4}{8}+\frac {x^2}{2}+1\right )+c_1 \left (\frac {8 x^6}{1155}+\frac {4 x^4}{77}+\frac {2 x^2}{7}+1\right ) x^{3/2} \]

✓ Sympy. Time used: 1.020 (sec). Leaf size: 46

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*y(x) + 2*x*Derivative(y(x), (x, 2)) - (2*x**2 + 1)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{6}}{48} + \frac {x^{4}}{8} + \frac {x^{2}}{2} + 1\right ) + C_{1} x^{\frac {3}{2}} \left (\frac {4 x^{4}}{77} + \frac {2 x^{2}}{7} + 1\right ) + O\left (x^{8}\right ) \]