Find y as a function of if 09y06y04 SolutionSubstituting emx

Find y as a function of if (0)-9,y(0)-6,y\"(0)-4

Solution

Substituting e^mx in the given diffential equation we get,

(m3 -12 m2 +27m)emx = 0

Therefore,

m3 -12 m2 +27m = 0

m(m2 -12 m +27) = 0

m(m-9)(m-3) = 0

Thus, m = 0, 3, 9

Thus general solution would be

y = A + Be^3x +Ce^9x

Initial conditions:

1) y(0) = 9

A + B + C = 9

2) y\'(0) = 6

y\'(x) = 3Be^3x + 9Ce^9x

Thus, y\'(0) = 3B + 9C = 6

3) y\"(0) = 4

y\"(x) = 9Be^3x + 81Ce^9x

Thus, y\"(0) = 9B + 81C = 4

Thus, we have 3 simultaneous equations that need to be solved

A + B + C = 9

3B + 9C = 6

9B + 81C = 4

Solving these equations we get

A = 175/27

B = 25/9

C = -7/27

Thus, the general solution is y = 175/27 + (25/9)e^3x - (7/27)Ce^9x