# What is a solution to the differential equation #dy/dt=e^t(y-1)^2#?

The General Solution is:

# y = 1-1/(e^t + C) #

We have:

We can collect terms for similar variables:

Which is a separable First Order Ordinary non-linear Differential Equation, so we can "separate the variables" to get:

Both integrals are those of standard functions, so we can use that knowledge to directly integrate:

Leading to the General Solution:

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This is a separable differential equation, which means it can be written in the form:

It can be solved by integrating both sides:

Now we can integrate both sides:

Resubstituting (and combining constants) gives:

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The solution to the given differential equation ( \frac{dy}{dt} = e^t(y-1)^2 ) is ( y(t) = 1 + \frac{1}{Ce^{-t} - 1} ), where ( C ) is an arbitrary constant.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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