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