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4*3*2*1=24
speed=dt
You need to clarify what you want to solve for. If you're solving for z, then we can say: dz/dt + 4et + z = 0 ∴ dz/dt = -4et - z ∴ ∫(dz/dt) dt = -2et2 - zt + C ∴ z = -2et2 - zt + C ∴ z + zt = -2et2 + C ∴ z(1 + t) = -2et2 + C ∴ z = (-2et2 + C) / (1 + t)
The question is to PROVE that dy/dx = (dy/dt)/(dx/dt). This follows from the chain rule (without getting into any heavy formalism). We know x and y are functions of t. Given an appropriate curve (we can integrate piece-wise if necessary), y can be written as a function of x where x is a function of t, i.e., y = y(x(t)). By the chain rule, we have dy/dt = dy/dx * dx/dt. For points where the derivative of x with respect to t does not vanish, we therefore have (dy/dt)/(dx/dt) = dy/dx.
Changing velocity and constant acceleration? Yes. Changing velocity indicates constant acceleration dv/dt = a constant(k) when v=kt. Then dv/dt= dkt/dt= k. the constant k can be positive , negative or zero.