Calculate y when dy/dx = 1/(bx+5)^3

dy = (bx+5)^-3 dx

let z = bx+5
then dz = b dx and dx = (1/b) dz

dy = (1/b) z^-3 dz

y = (1/b)(-1/2)z^-2 + c

y = -1/(2b z^2) + c
but z^2 = b^2x^2 + 10 bx + 25
so
2 b z^2 = 2b^3x^2 + 20 b^2 x + 50 b
so
y = -1/(2b^3x^2 + 20 b^2 x + 50 b) + c

or, given the original form of the function,

y = -1/(2(bx+5)^2)

I have a extra b in there.

actually, I dropped a b!

y = -1/(2b(bx+5)^2)

Two b or not two b

To find y when dy/dx is given as 1/(bx+5)^3, you need to solve the differential equation by integrating and applying initial conditions. Here's the step-by-step process:

Step 1: Start by separating the variables. The given differential equation can be rewritten as:

dy = (1/(bx+5)^3) dx

Step 2: Integrate both sides of the equation. The integral of dy is simply y, and the integral on the right-hand side, assuming b is a constant, can be evaluated using the power rule for integration:

y = ∫ (1/(bx+5)^3) dx

Step 3: To integrate 1/(bx+5)^3, make a substitution using u = bx+5. This will simplify the integral and make it easier to solve.

Let's calculate the integral of 1/u^3 with respect to u:

∫ (1/u^3) du = -1/(2u^2) + C,

where C is the constant of integration.

Step 4: Now, we have:

y = -1/(2u^2) + C

But remember, our original substitution was u = bx+5. Let's put u back in terms of x:

y = -1/(2(bx+5)^2) + C

That's the general solution to the differential equation. However, we need to apply an initial condition to find the particular solution.

If you have a specific x and y value that satisfies the equation, you can substitute those values to find the constant C. This will give you the particular solution to the differential equation.