54.3.14 problem 14

Internal problem ID [8570]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 17. Power series solutions. 17.5. Solutions Near an Ordinary Point. Exercises page 355
Problem number : 14
Date solved : Wednesday, March 05, 2025 at 06:09:47 AM
CAS classification : [_Gegenbauer]

\begin{align*} \left (-4 x^{2}+1\right ) y^{\prime \prime }+6 x y^{\prime }-4 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 39

Order:=8; 
ode:=(-4*x^2+1)*diff(diff(y(x),x),x)+6*x*diff(y(x),x)-4*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (2 x^{2}+1\right ) y \left (0\right )+\left (x -\frac {1}{3} x^{3}-\frac {1}{6} x^{5}-\frac {3}{14} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 40

ode=(1-4*x^2)*D[y[x],{x,2}]+6*x*D[y[x],x]-4*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_1 \left (2 x^2+1\right )+c_2 \left (-\frac {3 x^7}{14}-\frac {x^5}{6}-\frac {x^3}{3}+x\right ) \]

✓ Sympy. Time used: 0.874 (sec). Leaf size: 29

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(6*x*Derivative(y(x), x) + (1 - 4*x**2)*Derivative(y(x), (x, 2)) - 4*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=8)
 

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