Equations of state for compressible substances My thermodynamics textbook gives these two equations of state for compressible substances:

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*Thermal equation of state: $$v(T,p)= v_0 (1 + \beta \ T - \kappa p)$$

*Caloric equation of state: $$u(T,v)= \int c_v(T)\ dT + \frac{(v -v_0)^2}{2\kappa v_0}$$
where $c_v$ is the heat capacity at constant volume, $\kappa$ is the compressibility and $\beta$ the thermal expansion coefficient.
I assume the the thermal equation of state just states the basic assumption of the underlaying model, namely that the volume changes linearly with changes in pressure and temperature.
Unfortunately, my book does not provide any details about the reasoning behind these equations. My question is how to derive these equations, especially equation 2.
 A: The key is to note the left-hand terms: $v(T,p)$ and $u(T,v)$. The presence of two variables reminds us that for a closed system, we can add energy in two ways: work or heat. The implication is that any state variable can be expressed in differential form using at most two other state variables.
We start accordingly by expanding $dv$ in the variables $T$ and $p$ as
$$dv(T,p)=\left(\frac{\partial v}{\partial T}\right)_pdT+\left(\frac{\partial v}{\partial p}\right)_Tdp;$$
$$dv(T,p)=v\beta(T)\,dT-v\kappa(T)\,dp;$$
$$\frac{dv(T,p)}{v}=\beta(T)\,dT-\kappa(T)\,dp,$$
where we've used the definitions of the material properties $\beta$ (the thermal expansion coefficient) and $\kappa$ (the compressibility). Assuming temperature-independent material properties; integrating from $v_0$ to $v$, $T_0=0$ to $T$, and $p_0=0$ to $p$; and assuming small changes in volume so that we can Taylor-expand $\ln \frac{v}{v_0}\approx\frac{v-v_0}{v_0}$, we obtain the first equation you provide.
For the second equation, we expand $du$ ($=T\,dS-p\,dv$ for a closed system) in $T$ and $v$ as
$$du(T,v)=\left(\frac{\partial u}{\partial T}\right)_vdT+\left(\frac{\partial u}{\partial v}\right)_Tdv;$$
$$du(T,v)=c_v(T)\,dT+\left[T\left(\frac{\partial S}{\partial v}\right)_T-p\right]dv;$$
$$du(T,v)=c_v(T)\,dT+\left(\frac{T\beta(T)}{\kappa(T)}-p\right)dv,$$
where we've used a Maxwell relation and the material property definitions again. Now assume temperature-independent $\beta$ and $\kappa$ again, use $\frac{\beta T}{\kappa}=\frac{1}{\kappa}\left(\frac{v-v_0}{v_0}\right)+p$ from the first equation, integrate from $v_0$ to $v$, and assume $u_0=0$ to obtain the second equation you provide.
