Question

- For a substance C, the time rate of conversion is proportional to the square of the amount x of unconverted substance. Let k be the numerical value of the constant of proportionality and let the amount of unconverted substance be x0 at time t=0. Determine x for all t (greater than or equal to) 0.

Answer 1

wait i am wrong...this is a differential equation problem huh?
\[\frac{dx}{dt} = kx^{2}\]

Answer 2

separate variables
\[\frac{dx}{x^{2}} = k dt\]
integrate both sides
\[-\frac{1}{x} = kt + C\]
solve for x
\[x = \frac{1}{C-kt}\]
find C using initial value...x(0) = x_0
\[x_0 = \frac{1}{C} \rightarrow C = \frac{1}{x_0}\]
after simplifying
\[x(t) = \frac{x_0}{1-k x_0 t}\]

Answer 3

yeah this is differential equation problems

Answer 4

is this a final answer

Answer 5

i believe so

Answer 6

were you able to understand my work

Answer 7

may be i understand but i confuse if it is the real answer to the question , determine the x,for all t (greater than or equal to 0)

Answer 8

x is a function of t.... so to determine x for all positive t values, we have to define the function x(t)

Answer 9

ok tnx fo your help,,

Answer 10

actually just noticed something...right now the function has asymptote at t = 1/kx_0 which means x is not defined for all positive "t" values
so just switch the asymptote to negative side
i should have assumed the constant was negative
\[C = -\frac{1}{x_0}\]
this changes the function to:
\[x(t) = \frac{-x_0}{1+k x_0 t}\]