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

Problem #14: Consider the following differential equation. [6 marks] (1 + 5x2)y – 8xy – 6y = 0 (a) If you were to look for

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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o ->0 Given differential equation (itsely! gay! by=0 At x=0 is ordinary point Y= not axta, x² tez x3 +94x4 + topx & 7 be Solcoefficient of x- 1. Mt2) Cnt 1) antz + 5 nch-han- - 8 n. an-bano M+2) Cnti) antz +5n-1) an - snan-banzo -an 64-2)(n+1) Ant2Substituting a193,94 in ea g=aota, x + 390) at +291) x² + (300) et e- y = 26C1+34 +374 + -->) +0,62+--) Required solution y

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