Find f(x) if y=f(x) satisfies dy/dt= 18yx^5 and they y-intercept of the curve y=x(f) is 5?

Question

Trying to do math homework but cant figure out how to do this question. Help Please

Answer 1

I think you mean dy/dx, not dy/dt.
OK, so we have dy/dx = 18yx^5.
Move all the terms involving y to one side, and the xs to the other side.
So (1/y) dy = 18x^5 dx.
Now we can integrate both sides:
integral of 1/y dy = integral of 18x^5 dx
ln|y| = 3x^6 + c
Now, to get rid of the ln take e of both sides:
y = e^(3x^6 + c). You can also write that as y = Ae^(3x^6), because e^c is just a constant - its more usual to write it this way.
Now substitute in the point - x = 0, y = 5 to get 5 = A, so A = 5.
Thus the equation is y = 5e^(3x^6).

Answer 2

Do you mean that dy/dx = 18yx^5 ?
And presumably the y-intercept is for y = f(x) not y = x(f)?

Answer 3

y=f(x)
dy/dt=18yx^5
dy=18(yx^5)dt
y is equal to the integral of (18yx^5)dt

or Y=18integral(yx^5)dt
as y= f(x) ; f(x)=18integral(yx^5)dt