# Math Help - Integrating a function of compressibility

1. ## Integrating a function of compressibility

Hi, i have a differential equation I am trying to integrate, but I do not know how to do it. The equation is:

dp = rho*g*dx

rho = rho_0 + (p(h)-p_0)/c^2

dp = (rho_0 + (p(h)-p_0)/c^2)*g*dh

This is a linear equation for the compressibility in a fluid. p(h) is the pressure at a given height below the surface of the fluid, p_0 is a reference pressure (above the fluid, atmospheric), rho_0 is the density of this fluid at that pressure, c is a constant and g is the gravity constant.

Anyone knows what the expression for p(h) will be?

2. Originally Posted by balloo
Hi, i have a differential equation I am trying to integrate, but I do not know how to do it. The equation is:

dp = rho*g*dx

rho = rho_0 + (p(h)-p_0)/c^2

dp = (rho_0 + (p(h)-p_0)/c^2)*g*dh

This is a linear equation for the compressibility in a fluid. p(h) is the pressure at a given height below the surface of the fluid, p_0 is a reference pressure (above the fluid, atmospheric), rho_0 is the density of this fluid at that pressure, c is a constant and g is the gravity constant.

Anyone knows what the expression for p(h) will be?
Is this part of a bigger question? Could you post the question exactly as it was?

3. No, not really. It is only this function I need to integrate for it self.

4. Originally Posted by balloo
Hi, i have a differential equation I am trying to integrate, but I do not know how to do it. The equation is:

dp = rho*g*dx

rho = rho_0 + (p(h)-p_0)/c^2

dp = (rho_0 + (p(h)-p_0)/c^2)*g*dh

This is a linear equation for the compressibility in a fluid. p(h) is the pressure at a given height below the surface of the fluid, p_0 is a reference pressure (above the fluid, atmospheric), rho_0 is the density of this fluid at that pressure, c is a constant and g is the gravity constant.

Anyone knows what the expression for p(h) will be?
$dp = g\bigg(\rho_0 + \frac{p(h)-p_0}{c^2}\bigg)dh$

$dp = \bigg(g \rho_0 + \frac{g}{c^2}(p(h)-p_0)\bigg)dh$

$dp = \bigg(g \rho_0 + \frac{g}{c^2}(p(h))-\frac{g p_0}{c^2}\bigg)dh$

$\int dp = \int \bigg(g \rho_0 + \frac{g}{c^2}(p(h))-\frac{g p_0}{c^2}\bigg)dh$

$\int dp = \int g \rho_0 dh + \int \frac{g}{c^2}(p(h))dh- \int \frac{g p_0}{c^2}dh$

$p = \rho_0 g h + \frac{g}{c^2}\int (p(h))dh- \frac{gh p_0}{c^2}$

$p = g h\bigg(\rho_0 - \frac{p_0}{c^2}\bigg) + \frac{g}{c^2}\int (p(h))dh$

Can't really do much more than this.