Question

Find the general (explicit) solution to the differential equation: dy dx = y4 cos x

Answer 1

Try this solution:dy/dx=y⁴cosx; y⁻⁴dy=cosxdx;  $\int\ {y^{-4}} \, dy= \int\ {cosx} \, dx;$$- \frac{1}{3y^3}=sinx+C; \ =\ \textgreater \ \ y^3=- \frac{1}{3sinx+C}$

Answer 2

In this problem, we want to solve a separable differential equation.

We will find the solution:

$y = (3*sin(x) - c)^{-1/3} = \frac{1}{\sqrt[3]{ (3*sin(x) - c)}}$

Let's see how to get that solution:

We start with the differential equation:

$\frac{dy}{dx} = y^4*cos(x)$

To solve this, we can move all the terms with "x" to the right and all the terms with "y" to the left, to get:

$\frac{dy}{y^4} = cos(x)*dx$

Now we integrate in both sides, so we get:

$\int \frac{dy}{y^4} = \int cos(x)*dx\\\\-(1/3)*y^{-3} + c = -sin(x)$

Where c is a constant of integration.

Now to get the explicit solution, we just isolate y in one side of the equation:

$y = (3*sin(x) - c)^{-1/3} = \frac{1}{\sqrt[3]{ (3*sin(x) - c)}}$

If you want to learn more, you can read:

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