Answer:
1) [tex] y =4x^2 +7[/tex]
2) [tex] y =7s^2 -3s^4 +181[/tex]
Step-by-step explanation:
Assuming that our function is [tex] y = f(x)[/tex] for the first case and [tex] y=f(s)[/tex] for the second case.
Part 1
We can rewrite the expression like this:
[tex] \frac{dy}{dx} =8x[/tex]
And we can reorder the terms like this:
[tex] dy = 8 x dx[/tex]
Now if we apply integral in both sides we got:
[tex] \int dy = 8 \int x dx[/tex]
And after do the integrals we got:
[tex] y = 4x^2 +c[/tex]
Now we can use the initial condition [tex] y(0) =7[/tex]
[tex] 7 = 4(0)^2 +c, c=7[/tex]
And the final solution would be:
[tex] y =4x^2 +7[/tex]
Part 2
We can rewrite the expression like this:
[tex] \frac{dy}{ds} =14s -12s^3[/tex]
And we can reorder the terms like this:
[tex] dy = 14s -12s^3 dx[/tex]
Now if we apply integral in both sides we got:
[tex] \int dy = \int 14s -12s^3 ds[/tex]
And after do the integrals we got:
[tex] y = 7s^2 -3s^4 +c[/tex]
Now we can use the initial condition [tex] y(3) =1[/tex]
[tex] 1 = 7(3)^2 -3(3)^4 +c, c=1-63+243=181[/tex]
And the final solution would be:
[tex] y =7s^2 -3s^4 +181[/tex]
