Suppose the Wronskian of ff and gg is W(t)=5e12tW(t)=5e12t, where f(t)=e14tf(t)=e14t. Find the general formula for g(t)g(t), writing C for the arbitrary constant. g(t)=g(t)= 512e−2t+Ce14t512e−2t+Ce14t 5 12e−2t+ C e14t.

g = (-5/16)e^(-2t) + Ce^(14t)

Step-by-step explanation:

Given f(t) and g(t).

The Wronskian of f(t) and g(t) is given as

W(t) = 5e^(12t)

If f(t) = e^(14t)

We need to find g(t).

First, Wronskian of f and g is the determinant

W(t) = W(f,g) = Det(f, g; f', g')

= fg' - f'g

But f = e^(14t) and W(t) = 5e^(12t)

So

5e^(12t) = e^(14t).g' - 14e^(14t).g

Multiplying through by e^(-14t)

5e^(-2t) = g' - 14g

Now, we have the first order ordinary differential equation

g' - 14g = 5e^(-2t)

Multiply through by the integrating factor

IF = e^(integral of -14dt)

= e^(-14t)

So, we have

e^(-14t)[g' - 14g] = 5e^(-2t)e^(-14t)

d[ge^(-14t)] = 5e^(-16t)

Integrate both sides

ge^(-14t) = (-5/16)e^(-16t) + C

Multiplying through by e^(14t)

g = (-5/16)e^(-2t) + Ce^(14t)

And this is what we want.

the simplified form is

step-by-step explanation:

we want to simplify the expression,

let us change the middle bar to a normal division sign to obtain,

we multiply the first fraction by the reciprocal of the second fraction to obtain,

we factor to obtain,

we cancel the common factors to get,

we simplify to get,

the correct answer is b

when rotating a figure 180 degrees, imagine that you are able to take the figure and turn it clockwise or counterclockwise around a center point. to rotate a figure 180 degrees, you will need to apply the rule (x, y) → (-x, -y). start by using a coordinate grid with coordinates for each vertex of the figure. the center point of the coordinate grid is located at (0, 0), which is what you will rotate the figure around. write down the original coordinates of the shape you are going to rotate. then, apply the rule. for example, coordinate (1, 2) becomes (-1, -2). another example would be, coordinate (-4, -2) becoming (4, 2). after you change each original coordinate to the rotated coordinate, you will draw your new figure.

step-by-step explanation: