I'm reading a textbook in which an equation is rearranged and I'm failing to see how they've done it. I've tried writing it down step by step in my notebook but can't come up with the right answer. It's frustrating because I'm trying to learn calculus and differential equations - something left out of my education. After reading, I understand the idea behind calculus and what is being done and what it means, but I'm bad and working individual examples out.
It's a differential equation with the output set to 0 to find the initial value of $x_0$.
$$r_0 \left(1-\frac{x_0}{k_0}\right) - d_0 = 0$$
Rearranges to $$x_0 = \frac{k_0}{r_0}\left(r_0 - d_0\right)$$
It's the part with the fraction in the brackets that's throwing me, I can sort of see where they got $(r_0 - d_0)$ from.
I think to start with I need to get rid of the brackets so as: $$ r_0 - \frac{r_0x_0}{r_0k_0} - d_0 = 0$$
Then I think the $r_0$ on the top and bottom cancel?
$$r_0 - \frac{x_0}{k_0}-d_0 = 0$$
But I'm really not sure.
Thanks,
Ben.
Not difficult to see
$$\begin{align} r_0\Big(1 - \frac{x_0}{k_0}\Big) - d_0 = 0 &\implies r_0 - \frac{r_0 \ x_0}{k_0} - d_0 = 0 \\ &\implies r_0 - d_0 = \frac{r_0 \ x_0}{k_0} \\ &\implies \frac{k_0}{r_0}\Big(r_0 - d_0 \Big) = x_0 \end{align} $$