Calculus Differential Equations

Question

If y0 satisfies the equations

dy/dx + 6xy^4 = 0, y(0) = 1/2

find the value of y0(2)

1. 44^-1/3
2. 64^1/4
3. 56^1/3
4. -32^1/4
5. -28^1/3

need help really bad i wanna learn how to do this ! :)

Answer 1

The differential equation given in the problem is separable, meaning we can get everything involving y on one side and everything involving x on the other. Once we separate the variables, then we can just integrate both sides.

dy/dx + 6xy^4 = 0

dy/dx = -6xy^4

y^-4 dy = -6x dx

now we integrate both sides, remembering the constants of integration, to get:

(-1/3)y^-3 + C1 = -3x^2 + C2

we can clean this up a bit:

y^-3 = 9x^2 + C

Now, we know that y(0)= 1/2; we can use this to find C.

(1/2)^-3 = C = 8

Now, we can plug in x=2 to get the final result:

y^-3 = 9*2^2 + 8 = 44

so y = 44^(-1/3)

I hope this helps