Question

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

Answer 1

1)
`y =4x^2 +7`

2)
`y =7s^2 -3s^4 +181`

Explanation:

Assuming that our function is
`y = f(x)` for the first case and
`y=f(s)` for the second case.

Part 1

We can rewrite the expression like this:

`(dy)/(dx) =8x`

And we can reorder the terms like this:

`dy = 8 x dx`

Now if we apply integral in both sides we got:

`\int dy = 8 \int x dx`

And after do the integrals we got:

`y = 4x^2 +c`

Now we can use the initial condition
`y(0) =7`

`7 = 4(0)^2 +c, c=7`

And the final solution would be:

`y =4x^2 +7`

Part 2

We can rewrite the expression like this:

`(dy)/(ds) =14s -12s^3`

And we can reorder the terms like this:

`dy = 14s -12s^3 dx`

Now if we apply integral in both sides we got:

`\int dy = \int 14s -12s^3 ds`

And after do the integrals we got:

`y = 7s^2 -3s^4 +c`

Now we can use the initial condition
`y(3) =1`

`1 = 7(3)^2 -3(3)^4 +c, c=1-63+243=181`

And the final solution would be:

`y =7s^2 -3s^4 +181`