Question

Consider the differential equation $\frac{\mathbf{d}\mathbf{p}}{\mathbf{d}\mathbf{t}}\mathbf{=}\mathit{f}\mathbf{\left(}\mathbf{P}\mathbf{\right)}$

Where $\mathit{f}\mathbf{\left(}\mathit{P}\mathbf{\right)}\mathbf{=}\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{5}{\mathit{P}}^{\mathbf{3}}\mathbf{-}\mathbf{1}\mathbf{.}\mathbf{7}\mathit{P}\mathbf{+}\mathbf{3}\mathbf{.}\mathbf{4}$

The function $\mathit{f}\mathbf{\left(}\mathit{P}\mathbf{\right)}$has one real zero, as shown in Figure 2.R.3. Without attempting to solve the differential equation, estimate the value of $\underset{\mathbf{t}\mathbf{\to }\mathbf{\infty }}{\mathbf{l}\mathbf{i}\mathbf{m}}\mathit{P}\mathbf{\left(}\mathit{t}\mathbf{\right)}$.

Short answer

The estimated value of $\underset{t\to \infty }{\lim }P\left(t\right)$is $1.32139$

Step-by-step solution

Define Newton method

Newton's Approach is an iterative method for solving the system of equations g(x)=0 with an approximate solution. As input, the technique requires a first guess X⁽⁰⁾ It then computes following iterates X⁽¹⁾ . X⁽²⁾ which should ideally converge to a solution x^* of g(x)=0.

Find the equation if they are separable or not.

We want to know for which the value of t the value of P does not change any longer,

i.e., we want to determine the P critical point.

As a result, we must discover the equation's solution.

$\frac{\mathrm{d}P}{\mathrm{d}t}=0$

Since $\frac{\mathrm{d}P}{\mathrm{d}t}=f\left(P\right)$

The equivalent the solution

$f\left(P\right)=0$

i.e., we need to find the roots of P

Fortunately, we have the graph of t

So we can deduce the answer by reading from the graph and approximating even further with the Newton method:

The value of$\underset{t\to \infty }{\lim }P\left(t\right)is1.32139$