What do I do with e^y in the differential equation dy/dt =(2t)/(e^y)?

As you said, the equation is separable:

dy/dt = 2t / exp(y)

exp(y) dy = 2t dt

Integrate both sides:

∫ exp(y) dy = ∫ 2t dt

exp(y) + C₁ = t ² + C₂

Move the constant terms to one side. When you add them together, you get another constant, so you can ignore the subscript altogether:

exp(y) = t ² + C

Solve for y explicitly by taking the logarithm of both sides:

ln(exp(y)) = ln(t ² + C )

y = ln(t ² + C )

C can be any number; if it happens to be 0, then you have

y = ln(t ²) = 2 ln(t )

so B is the correct choice.

You can also approach this from the opposite angle: Assume y is one of the given solutions, then substitute it into the ODE. (Bit more trial-and-error involved, so not a good idea if you're short on time.)

For example, if y = 2 exp(t ) as in choice A, you have dy/dt = 2 exp(t ), so the ODE would become

2 exp(t ) = 2t / exp(2 exp(t ))

which is clearly (I hope) not true.

4

step-by-step explanation:

step-by-step explanation:

we are given a graph for system of inequalities.

in order to write the system of inequalities for the given graph, we need to find the inequality for each of the line separately.

let us find the slope of dotted line first.

taking two coordinates on the graph.

let us say (0,0) and (3,1).

let us find the slope of the line now.

slope of the dotted line would be =.

y-intercept of dotted line is y=0 only.

plugging slope m= and b=0 in slope-intercept form y=mx+b, we get

.

the line is shaded up of that dotted line.

therefore, for shading up we would take greater than inequality symbol > .

and because it's a dotted line we would take only > symbol in place of = sign in above equation of line.

therefore, inequality for dotted line would be

let us take two points (3,1) and (0,4) on the solid line.

slope of the solid line would be =.

for the solid line we have y-intercept b= 4.

plugging slope m= -1 and b=4 in slope-intercept form y=mx+b, we get

y = -1x + 4.

the line is shaded down of that solid line.

therefore, for shading down we would take less than inequality symbol < .

and because it's a solid line we would take ≤ symbol in place of = sign in above equation of line.

therefore, inequality for solid line would be