Determine a lower bound for the radius of convergence of series solutions for the differential equation, (x2 – 3x – 54)y"+ xy + 4y = 0, %3D about the points xo = 4, x = -4, and xo = 0. NOTE: Enter o if the series solutions converge everywhere. For co 4, Pmin %3D For xo = -4, Pmin For xo = 0, Pmin

Find the first four nonzero terms in a power series expansion about x = 0 for a general solution to the given differential equation. (x² +21)y"+y=0 y(x) =+ (Type an expression in terms of a and a, that includes all terms up to order 3.)

The point x = O is a regular singular point of the given differential equation. Find the recursive relation for the series solution of the DE below. Show the substitution and all the steps to obtain the recursive relation. Do not solve the equation for y=y(x) xy" + 4y' - xy = 0, 1 a. Ck+1= (k+r+ 1)(k+r+3) 1 b. Ck+1 = (k+r+1)(k+r+4) ·Ck-1, k≥ 1 с. Ск d. Ck² e. Ck O a P = e 11 || 1 k+r I k+r ·Ck-1, k≥1 -Ck-1, k≥1 -Ck-1, k≥1 (k+r)² +5(k+r) 1 (k+r)²-2(k+r) -8 ·CK-2, k≥2

Try to use the method of Frobenius to find a series expansion about the irregular singular point x = 0 for a solution to the given differential equation. If the method works, give the first four nonzero terms in the expansion. If the method does not work, explain what went wrong. 3x²y+3y' - 6y=0