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

##### 2 Answers

The General Solution is:

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

#### Explanation:

We have:

# dy/dt = e^t(y-1)^2 #

We can collect terms for similar variables:

# 1/(y-1)^2 \ dy/dt = e^t #

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

# int \ 1/(y-1)^2 \ dy = int e^t \ dt #

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

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

And we can readily rearrange for

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

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

Leading to the General Solution:

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

#### Explanation:

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

It can be solved by integrating both sides:

In our case, we first need to separate the integral into the right form. We can do this by dividing both sides by

Now we can integrate both sides:

We can solve the left hand integral with a substitution of

Resubstituting (and combining constants) gives:

Multiply both sides by

Divide both sides by