Derive equation with a multiplier and constant

[TMonkey]

Hi there,

I'm trying to derive the following equation, but despite internet searching haven't been able to figure out what to do with the multiplier, D.

D * dθ/dz + k = C

I need to derive the equation and then plot the graph θ(y-axis) against z (x-axis).

Any help greatly appreciated,
thank you.

 

[vanhees71]

Deriving from what? It helps a lot to clearly state the problem not only to be able to get help from others but also to solve the problem yourself!

 

[UltrafastPED]

Derive the equation from what? Is it from raw data, or from a description of a property of something?

You would do better to look through your assignment and class notes rather than the internet! Or form a study group.

 

[HallsofIvy]

If by "derive the equation" you mean solve that equation for [itex]\theta[/itex] as a function of z just solve for [itex]d\theta= [(C- k)/D]dz[/itex]. If C, k, and D are constants that gives [itex]\theta= [(C- k)/D]z+ E[/itex] where E is a "constant of integration".

 

[blue_raver22]

Sure, I'd be happy to help with deriving this equation. First, let's rewrite the equation with the multiplier and constant separated from the derivative term:

D * dθ/dz = C - k

To derive this equation, we can use the standard rules for differentiating a product of two functions. In this case, the two functions are D and dθ/dz. The rule states that the derivative of a product is equal to the first function times the derivative of the second function, plus the second function times the derivative of the first function. In mathematical notation, this can be written as:

d/dz (D * dθ/dz) = D * d^2θ/dz^2 + dD/dz * dθ/dz

Now, we can substitute this into our original equation:

D * d^2θ/dz^2 + dD/dz * dθ/dz = C - k

Next, we can rearrange the equation to solve for the second derivative of θ with respect to z:

d^2θ/dz^2 = (C - k - dD/dz * dθ/dz) / D

Finally, to plot the graph of θ against z, we can integrate both sides of the equation with respect to z. This will give us an expression for θ in terms of z. We can then use this expression to plot the graph. I hope this helps! Let me know if you have any further questions.