Find the particular solution that satisfies the differential equation and the initial condition.

1. f '(x) = 8x, f(0) = 7

2. f '(s) = 14s − 12s3, f(3) = 1

1)

2)

Step-by-step explanation:

Assuming that our function is for the first case and for the second case.

Part 1

We can rewrite the expression like this:

And we can reorder the terms like this:

Now if we apply integral in both sides we got:

And after do the integrals we got:

Now we can use the initial condition

And the final solution would be:

Part 2

We can rewrite the expression like this:

And we can reorder the terms like this:

Now if we apply integral in both sides we got:

And after do the integrals we got:

Now we can use the initial condition

And the final solution would be:

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step-by-step explanation: