The limit of case 2

Up a level : Differential Equations
Previous page : Brine mixture, case 2, inflow≠outflow, pure water in
Next page : Brine mixture, case 3, inflow=outflow, brine in

The limit of case 2

We have two rather algebraically differently looking solutions, an exponential but as r1 → r2 the limit of the second case ought to be the first case. Before you read any further, why not try to figure it out yourself?

*

*

*

*

*

*

*

*

*

*

Ok, back again. Let us have a look at

y = {y_0}{\left( {1 - \frac{{\left( {{r_2} - {r_1}} \right)}}{{{V_0}}}t} \right)^{\frac{{{r_2}}}{{{r_2} - {r_1}}}}}

Now let us do the substitution

n = \frac{{{r_2}}}{{{r_2} - {r_1}}}

That means that

\frac{{\left( {{r_2} - {r_1}} \right)}}{{{V_0}}} = \frac{{{r_2}/{V_0}}}{n}

and thus that

y = {y_0}{\left( {1 - \frac{{{r_2}/{V_0}}}{n}t} \right)^n}

But we know that

{e^x} = \mathop {\lim }\limits_{n \to \infty } {\left( {1 + \frac{x}{n}} \right)^n}

and thus that

y = {y_0}{\left( {1 - \frac{{{r_2}/{V_0}}}{n}t} \right)^n} = {y_0}{e^{ - \frac{{{r_2}}}{V}t}} = {y_0}{e^{ - \frac{r}{V}t}}

I.e. the solution we found in case 1.

Up a level : Differential Equations
Previous page : Brine mixture, case 2, inflow≠outflow, pure water in
Next page : Brine mixture, case 3, inflow=outflow, brine inLast modified: Dec 28, 2023 @ 17:28