Consider a differential equation $\frac{dp}{dv}=f(v,p)$ and solving the differential equation yield
$g(v,p)=$ constant and initial condition $p(v_0)=p_0$ implies $g(p_0,v_0)$=constant so i use this equation to find the value of p at point $(v_0+h)$ or equivalently $g(p_0,v_0)=g(p(v_0+h),v_0+h)$. In mathematical how do we adress this "$p(v_0+h)$"? The solution of p that satisfy the solution of the DE $\frac{dp}{dv}=f$ at point $v_0+h$?
Example
$\frac{dp}{dv}=f(v,p)=-1.4\frac{p}{v}$
Solve the DE and get
$pv^{1.4}=constant=g(p,v)$
$p(v_0)=p_0$ then $p(v_0+h)$=$\frac{p_0v_0^{1.4}}{(v_0+h)^{1.4}}$