The isothermal bulk modulus, *K*_{T}, is assumed to be independent from compression, namely pressure in the case of the simplest EOS. Here, we assume that the isothermal bulk modulus is a linear function of pressure. Namely,

(1)

where *K*_{T0} is the isothermal bulk modulus at zero pressure, and *K*_{0}' is its first pressure derivative.

By generalizing Eq (2) in the simplest EOS page, we have:

(2)

(3)

To solve the differential equation of Eq (3),

(4)

By integrating the both sides of Eq (4),

(5)

The integral constant *C* is determined from the conditions of *V* = *V*_{0} at *P* = 0:

(6)

By substituting Eq. (6) to Eq. (5), we have:

(7)

By taking out the natural logarithms, we have:

(8)

Finally, we have Murnaghan's integrated linear EOS:

(9)

The Murnaghan's EOS has been much less often used than the 3^{rd}-order Burch-Murnaghan EOS. The 3^{rd}-order Burch-Murnaghan EOS seems to be considered more advanced than the Murnaghan's EOS. However, as is discussed in the EOS page, the Murnaghan's EOS also shows satisfactory results at least for MgSiO_{3} bridgmanite. Therefore, the Murnaghan's EOS is equivalently
useful to other EOS for researchers in physics of the Earth and planetary interiors.