Question

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

Answer 1

Answer:

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

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

Step-by-step 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:

$\frac{dy}{dx} =8x$

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:

$\frac{dy}{ds} =14s -12s^3$

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:

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