Find the function y1 of t which is the solution of

25y−30y+8y=0
with initial conditions y1(0)=1y1(0)=0
y1=

Find the function y2 of t which is the solutionof
25y−30y+8y=0

with initial conditions y2(0)=0y2(0)=1
y2=

Find the Wronskian
W(t)=W(y1y2)

W(t)=

a) y₁(t) = (2.5e⁰•⁸ᵗ - 2.5e⁰•⁴ᵗ)

b) y₂(t) = (2e⁰•⁴ᵗ - e⁰•⁸ᵗ)

c) W(y₁,y₂) = -e¹•²ᵗ

Step-by-step explanation:

To solve, 25y" - 30y' + 8y = 0

With conditions, y₁'(0) = 1 and y₁(0) = 0 for y₁

And y₂'(0) = 0 and y₂(0) = 1 for y₂

Solving for y₁ first, using Laplace transforms.

Note:

L(y") = s² Y(s) - sy(0) - y'(0)

L(y') = s Y(s) - y(0)

L(y₁") = s² Y(s) - sy₁(0) - y'₁(0) = s² Y(s) - 0 - 1 = s² Y(s) - 1

L(y₁') = s Y(s) - y₁(0) = s Y(s)

L [25y" - 30y' + 8y] = L (0)

25(s²Y(s) - 1) - 30(sY(s)) + 8Y(s) = 0

25s²Y(s) - 25 - 30sY(s) + 8Y(s) = 0

Y(s) [25s² - 30s + 8] - 25 = 0

Y(s) [25s² - 30s + 8] = 25

Y(s) = (25)/[25s² - 30s + 8]

Y(s) = (25)/[(5s-4)(5s-2)]

To continue, we need to resolve 1/[(5s-4)(5s-2)] into partial fractions.

The calculation is continued on the attached images.

a.   since the profit made in 2012 is equal to the profit made in 2014, the profit is symmetric about the line y = 120,000.

step-by-step explanation:

b

step-by-step explanation: