Question

In Example 3 the solution $\mathit{y}\left(x\right)$ becomes unbounded as$\mathit{x}\mathbf{\to }\mathbf{\pm }\mathbf{\infty }$nevertheless,$\mathit{y}\left(x\right)$ is asymptotic to a curve as$\mathit{x}\mathbf{\to }\mathbf{-}\mathbf{\infty }$ and to a different curve as$\mathit{x}\mathbf{\to }\mathbf{\infty }$.What are the equations of these curves?

Short answer

The answer is: $y\left(x\right)$is asymptotic to the curve$y=2x+3.$when$x\to -\infty$and to the curve $y=2x-3$when $x\to \infty$.

Step-by-step solution

Given Information

The solution in example 3 was

$y=2x+\frac{3\left(1-e^{6x}\right)}{1+e^{6x}}$

which clearly becomes unbounded as$x\to \pm \infty .$

Rewrite the given equation

As$x\to -\infty ,e^{6x}\to 0$, the solution is close to$y=2x+3.$

$\begin{array}{c}y=2x+\frac{3\left(1-e^{6x}\right)}{1+e^{6x}}\\ =2x+\frac{3\left(1-e^{6x}\right)}{1+e^{6x}}·\frac{e^{-6x}}{e^{-6x}}\\ =2x+3\frac{e^{-6x}-1}{e^{-6x}+1}\end{array}$

Now, as $x\to \infty ,e^{-6x}\to 0$and therefore, the solution is close to$y=2x+3.$

Therefore, $y\left(x\right)$is asymptotic to the curve$y=2x+3.$ when $x\to -\infty$and to the curve$y=2x-3$ when $x\to \infty$.