Step by step
The question
Question: solve dy/dx = e^(2x).
dydx = e^(2x)
1 The idea
dy/dx is only about x.
So y is a function whose derivative is e^(2x): an antiderivative.
y′ = e^(2x): y is a function whose derivative is e^(2x)
2 Integrate both sides
Integrate both sides.
The left side, the integral of dy/dx, is just y.
y = ∫ e^(2x) dx
The rule: divide by the number in front of x.
∫ ekx dx = ekxk
Each term by its rule, and one + C: the number we cannot see after a derivative.
y = e^(2x)2 + C
Tidy the numbers: y = ½e^(2x) + C.
y = 12e^(2x) + C
Careful!
Don't forget + C.
Every number C gives another solution.
Careful: not y = 12e^(2x) ✗
The answer: y = ½e^(2x) + C.
C can be any number: one solution for each C.
y = 12e^(2x) + C
3 Check
Let's check: the derivative of our answer (the C goes, its derivative is 0) is e^(2x).
The right side again.
Right!
y′ = e^(2x)
Let's look back.
First: the idea.
Then: integrate both sides.
Last: check.
Answer
y = 12e^(2x) + C