Is it ok to reduce this equation?

Question

I have the next equation and it is necessary to integrate both sides: $$\dfrac{d(x_0x_1)}{dz}=-8\dfrac{dx_0}{dz}(r_i-z(r_i-1))$$ Is it ok to reduce this equation with $dz$, because of term $r_i-z(r_i-1)$, but $dz$ is on both sides? In that case I would get this equation: $$d(x_0x_1)=-8dx_0(r_i-z(r_i-1))$$

Answer 1

$$\dfrac{d(x_0x_1)}{dz}=-8\dfrac{dx_0}{dz}(r_i-z(r_i-1))$$ This symbolism supposes that $x_0$ and $x_1$ are some functions of $z$, namely $x_0(z)$ and $x_1(z)$. $$d(x_0x_1)=-8(r_i-z(r_i-1))dx_0\quad\text{is correct.}$$ But it is of no use to integrate both sides because the right side is not a total differential. $$x_0x_1=-8(r_i-z(r_i-1))x_0+C\quad\text{is not correct.}$$ One cay write :$\quad d(x_0x_1)=-8r_i dx_0+8(r_i-1)zdx_0$

$$x_0x_1=-8r_ix_0+8(r_i-1)\int z\:dx_0$$ where $z$ is function of $x_0$, namely $z(x_0)$ , in other words the inverse function of $x_0(z)$.

Or, if you prefer : $$x_0x_1=-8r_ix_0+8(r_i-1)\int z\left(\frac{dx_0}{dz}\right)dz$$

The above integrals cannot be evaluated or simplified or put on closed form without more information about the function $x_0(z)$ .