Differential equation dx/dt=-k(a-x)

Given the equation:
$$\frac{d}{d t} x{\left(t \right)} = - k \left(a - x{\left(t \right)}\right)$$
This differential equation has the form:
$$f_1(x)*g_1(x)*x' = f_2(x)*g_2(x)$$
where
$$f_{1}{\left(t \right)} = 1$$
$$g_{1}{\left(x \right)} = 1$$
$$f_{2}{\left(t \right)} = - k$$
$$g_{2}{\left(x \right)} = a - x{\left(t \right)}$$
We give the equation to the form:
$$\frac{g_1(x)}{g_2(x)}*x'= \frac{f_2(x)}{f_1(x)}$$
Divide both parts of the equation by $g_{2}{\left(x{\left(t \right)} \right)}$
$$a - x{\left(t \right)}$$
we get
$$\frac{\frac{d}{d t} x{\left(t \right)}}{a - x{\left(t \right)}} = - k$$
We separated the variables t and x.

Now, multiply the both equation sides by dt,
then the equation will be the
$$\frac{dt \frac{d}{d t} x{\left(t \right)}}{a - x{\left(t \right)}} = - dt k$$
or
$$\frac{dx}{a - x{\left(t \right)}} = - dt k$$

Take the integrals from the both equation sides:
- the integral of the left side by x,
- the integral of the right side by t.
$$\int \frac{1}{a - x}\, dx = \int \left(- k\right)\, dt$$
Detailed solution of the integral with x
Detailed solution of the integral with t
Take this integrals
$$- \log{\left(a - x \right)} = - k t + Const$$
Detailed solution of the equation
We get the simple equation with the unknown variable x.
(Const - it is a constant)

Solution is:
$$x_{1} = x{\left(t \right)} = C_{1} e^{k t} + a$$