Question

Question:

(a) Solve the DE in Problem 28 subject toy(1) = 0. For convenience let $\mathbf{K}\mathbf{}\mathbf{=}{\mathbf{v}}_{\mathbf{r}}\mathbf{/}{\mathbf{v}}_{\mathbf{s}}$
(b) Determine the values offor which the swimmer will reach the pointby examining$\mathbf{〖}\mathbf{lim}{\mathbf{〗}}_{\mathbf{(}}\mathbf{x}\mathbf{\to }\mathbf{0}\mathbf{)}\mathbf{ }\mathbf{y}\mathbf{(}\mathbf{x}\mathbf{)}$in the cases $\mathbf{k}\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{k}\mathbf{>}\mathbf{1}\mathbf{,}$and $\mathbf{0}\mathbf{<}\mathbf{k}\mathbf{<}\mathbf{1}\mathbf{.}$

Short answer

(a)

The differential equation that models the position of the swimmer across the river is

with initial condition y(1) = 0 . We are also given that $\mathbf{K}\mathbf{=}{\mathbf{\nu }}_{\mathbf{r}}\mathbf{/}{\mathbf{\nu }}_{\mathbf{s}}$.

We simplify the differential equation by rewriting the fraction on the right-hand side and substituting K.

Step-by-step solution

Write the expression of differential equation

(a)

The differential equation that models the position of the swimmer across the river is

with initial condition y(1) = 0 . We are also given that $\mathbf{K}\mathbf{=}{\mathbf{\nu }}_{\mathbf{r}}\mathbf{/}{\mathbf{\nu }}_{\mathbf{s}}$.

We simplify the differential equation by rewriting the fraction on the right-hand side and substituting K.

Take first-order differential equation

We have a first-order differential equation.

Recall that such an equation is homogeneous if f (tx, ty) = $t^{\alpha }f\left(x,y\right)$for some real integer $\alpha$.

So, we have a first-order homogeneous differential equation.

To solve a first-order homogenous differential equation we let y = mx.

Therefore, we havedy = ydc+ wch. Substituting these expressions into the differential equation yields,

Solve by separation of variables

We can solve the differential equation by separation of variables.

The integral on the left-hand side can be solved using -substitution. To avoid using the same variable twice we will use instead of for the substitution. We will use the substitution

Using the property of logarithms to eliminate the logarithms after integrating.

Writing the solution back in terms of just x and y using the substitution y=m from before yields,

Simplify the value of ‘y’

Now we can solve for the constant ${\mathit{c}}_{\mathbf{2}}$ using the initial condition y (1) = 0.

The solution to the differential equation becomes

Finally, we want to write the solution in terms of just x.

First, we isolate and eliminate the radical.

Now we can isolate y.

Recall values as given part (a)

(b)

Taking the limit, we have

Take Limit

Final calculation