The pressure, p, of the atmosphere at an altitude z is given...

The pressure, $p$, of the atmosphere at an altitude $z$ is given by $$ \frac{\mathrm{d} p}{\mathrm{~d} z}=-k p^{\frac{1}{\gamma}}(k \neq 0) $$ where $\gamma$ is the specific heat constant $(\gamma>1)$ and $k$ is a constant. Show that $$ z=\frac{\gamma p^{\frac{\gamma-1}{\gamma}}}{k(1-\gamma)}+A $$ (where $A$ is a constant).

The pressure, $p$, of the atmosphere at an altitude $z$ is given by
$$
\frac{\mathrm{d} p}{\mathrm{~d} z}=-k p^{\frac{1}{\gamma}}(k \neq 0)
$$
where $\gamma$ is the specific heat constant $(\gamma>1)$ and $k$ is a constant. Show that
$$
z=\frac{\gamma p^{\frac{\gamma-1}{\gamma}}}{k(1-\gamma)}+A
$$
(where $A$ is a constant).

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Key Concepts

Power Rule for Integration

The integration method used here relies on the power rule, which is applicable when integrating monomials of the form p^?. Once the differential equation is separated, integrating an expression like dp divided by p^(1/?) requires applying the power rule, yielding an expression involving p raised to a power that is adjusted by the constant exponent, a crucial step in reaching the final solution.

Constant of Integration

When integrating an indefinite integral, an arbitrary constant (often denoted as A or C) is introduced. This constant represents the family of solutions that differ by a constant shift and must be determined by initial or boundary conditions if provided. In the context of the question, including this constant is vital to expressing the general solution for the altitude z.

Separable Differential Equations

This concept involves differential equations that can be arranged so that all terms involving one variable (and its differential) are on one side of the equation and all terms involving the other variable are on the opposite side. In the question, the equation is separable since dp/dz can be written as a function of p multiplied by a constant, allowing one to separate variables and integrate both sides to obtain an antiderivative.

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