Mister Exam

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

y() =
y'() =
y''() =
y'''() =
y''''() =

from to

### The solution

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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:
$$x_{1} = x{\left(t \right)} = C_{1} e^{k t} + a$$
               k*t
x(t) = a + C1*e   
$$x{\left(t \right)} = C_{1} e^{k t} + a$$
$$x\left(t\right)=\left(x\left(0\right)-a\right)\,e^{k\,t}+a$$
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
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