Find y as a function of t if y" + 14y' + 85y = 0, y(0) = 9, y'(0) = 8. y(t) =

Step-by-step explanation:

The Given diffrential equation can be written as follows

Solving the auxiliary equation we get

Comparing with the standard quadratic equation we get

Using the values we get the roots as

The roots are imaginary and conjugate

For a auxillary equation with roots the solution is given by

Applying values we get the solution of the equation as

....................(i)

here are constants whose values shall be obtained using the given boundary conditions y(0) = 9 ; y'(0) = 8

In equation i putting x = 0 we get

9 =

Now we have

Now y'(0) = 8

Thus we get

Solving for we get

Thus the solution becomes

1 number

if we use a graphing calculator, there is only one such number that satisfies the conclusion of the mean value theorem, that is, a number such that

where the right-hand side is the average rate of change of f on the interval [0.5,0.75], which is the slope of the secant line.

in the attached graph, the black line is the secant line which has its slope as the average rate of change of f on [0.5,0.75], while the orange line is the only tangent line to the graph of f in the interval that satisfies the mvt conclusion.