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d --(x(t)) = -k*(a - x(t)) dt

$$\frac{d}{d t} x{\left(t \right)} = - k \left(a - x{\left(t \right)}\right)$$

x' = -k*(a - x)

Detail solution

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:

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

$$\frac{d}{d t} x{\left(t \right)} = - k \left(a - x{\left(t \right)}\right)$$

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:

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

The answer (#2)
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$$x\left(t\right)=\left(x\left(0\right)-a\right)\,e^{k\,t}+a$$

x = (x(0)-a)*E^(k*t)+a

The classification

1st exact

1st exact Integral

1st linear

1st linear Integral

1st power series

Bernoulli

Bernoulli Integral

almost linear

almost linear Integral

factorable

lie group

nth linear constant coeff undetermined coefficients

nth linear constant coeff variation of parameters

nth linear constant coeff variation of parameters Integral

separable

separable Integral