Step by step
The question
Question: solve dy/dx = 4y, with y(0) = −2.
The x parts on one side, the y parts on the other.
y(0) = −2 is the starting value: when x is 0, y is −2.
The right side splits into an x part, 4, times a y part, y.
That is a separable equation.
Our goal: each side with only one letter, so we can integrate each side.
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Separate x and y
Divide both sides by y, so y is only on the left.
Now multiply both sides by dx.
All the y parts are with dy, all the x parts with dx.
2 Integrate both sides
Put an integral sign on both sides.
On the left: ln |y|.
On the right: 4x.
Each side would get its own number; the two join into one, so we write one + C.
The integral of 1/y is the natural log of the size of y.
3 Solve for y
To undo ln, use e on both sides: e and ln undo each other, so e to the ln of |y| is just |y|.
A plus in the power splits into a times: e^(4x + C) = e^C × e⁴ˣ.
Careful!
The C is not added at the end.
e^C is a number that multiplies.
e^C is just some number.
With the ± from |y|, call it A: any number.
4 Use the starting value
Put in x = 0 and y = −2.
The power is 0 there, and e⁰ = 1.
So A = −2.
So y = −2e⁴ˣ.
5 Check
Let's check with a calculator.
At x = 1, our answer gives y ≈ −109.196.
The slope between two very close points: about −436.79.
The right side at that point: about −436.79.
The same slope.
Right!
6 Pick the choice
Our answer is y = −2e⁴ˣ.
That is choice C.
7 Why not the others
A is −2eˣ.
Wrong: lost the 4.
B is e⁴ˣ − 3.
Wrong: added the constant after taking e: it multiplies.
D is 2e⁴ˣ.
Wrong: lost the minus of y(0) = −2.
Let's look back.
First: separate x and y.
Then: integrate both sides.
Then: solve for y.
Then: use the starting value.
Then: check.
Then: pick the choice.
Last: why not the others.