# Consider the following differential equation. (1 + 5x2) y′′ − 8xy′ − 6y  =  0 (a)...

 Consider the following differential equation. (1 + 5x2) y′′ − 8xy′ − 6y  =  0
(a) If you were to look for a power series solution about x0  =  0, i.e., of the form
 ∞ Σ n=0
cnxn

then the recurrence formula for the coefficients would be given by ck+2  =  g(k) ck , k  ≥  2. Enter the function g(k) into the answer box below.
 (b) Find the solution to the above differential equation with initial conditions y(0)  =  0 and y′(0)  =  9. (Note that this solution is a terminating power series.)
 (c) Find the first three nonzero terms in the solution to the above differential equation with initial conditions y(0)  =  7 and y′(0)  =  0.
Problem #14: Consider the following differential equation. [6 marks] (1 + 5x2)y" – 8xy' – 6y = 0 (a) If you were to look for a power series solution about xo = 0, i.e., of the form Σc, Cnxh n=0 then the recurrence formula for the coefficients would be given by Ck+2 = g(k) Ck, k > 2. Enter the function gk) into the answer box below. (b) Find the solution to the above differential equation with initial conditions y(0) = 0 and y'(0) = 9. (Note that this solution is a terminating power series.) (c) Find the first three nonzero terms in the solution to the above differential equation with initial conditions y(O) = 7 and y'(0) = 0. (-5*k*(k-1)+8*+6)/((k+2)*(k+1)) Enter your answer as a symbolic function of k, as in these examples Problem #14(a): -5k(k – 1) + 8 + 6 (k + 2)(k + 1) 9*x+36*x^3 Enter your answer as a symbolic function of x, as in these examples Problem #14(b): 9x + 3683 7+7*(6/2)*x^2+7 Enter your answer as a symbolic function of X, as in these examples Problem #14(c): 7+7Qx2 +7

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