Respuesta :

cdw2014
Separate the x's and y's.

dy/y = x^7 dx

Integrate both sides.

ln(abs(y)) = (x^8)/8 + C

To cancel the natural root, make both sides the power to e.

e^ln(abs(y)) = e^((x^8)/8 + C)

abs(y) = e^C * e^((x^8)/8)

y = + or - [e^C * e^((x^8)/8)

Now just bundle the + or - e^C into a single constant. We will call it A.

y = Ae^((x^8)/8)

Now plug in the point (1,3).

3 = Ae^((1^8)/8)

3 = Ae^(1/8)

A = 3/(e^(1/8))

So the equation is:

y = (3/(e^(1/8))*(e^((x^8)/8)






facundo3141592

By separating the differential equation, we will see that the solution is:

y = exp( (1/7)*x^7 + 0.96)

How to solve a separable ODE?

Here we have the differential equation:

dy/dx = x^6*y

What we need to do, is move "separate" the variables, we will get:

dy/y = dx*x^6

Now we integrate both sides.

integrating that we will get:

ln(y)  = (1/7)*x^7  + C

We wrote the two constants of integration into only one.

Now we use the initial condition that the equation passes through the point (1, 3).

So if we evaluate x in 1 and y in 3, we should have the same value in each side of the equation:

ln(3) = (1/7)*1^7 + C

C = Ln(3) - 1/7 = 0.96

Then we have:

ln(y) = (1/7)*x^7 + 0.96

Now we just solve for y:

y = exp( (1/7)*x^7 + 0.96)

If you want to learn more about differential equations, you can read:

https://brainly.com/question/18760518

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