Q. 15 - Calculus

Question

 Explain how the equality ${\left.\frac{dy}{dx}\right|}_{\left(x_k,y_k\right)}=\frac{\Delta y_k}{\Delta x}$ is relevant to Euler's method.

Step-by-Step Solution

Answer

The iterative formula for determining the sequence of points required for applying Euler's technique to obtain an approximate solution to an initial-value issue uses the formula ${\left.\frac{dy}{dx}\right|}_{\left(x_i,y_2\right)}=\frac{\Delta y_k}{\Delta x}$

Step 1: Given Information

The equality ${\left.\frac{dy}{dx}\right|}_{\left(x_k,y_k\right)}=\frac{\Delta yk}{\Delta x}$ is relevant to Euler's method.

Step 2: Calculation

The iterative formula produces the sequence of points $\left(x_k,y_k\right)$ for the initial value issue specified by$\frac{dy}{dx}=g(x,y),y\left(x_0\right)=y_0$  and$\Delta x>0$.

$\left(x_{k+1},y_{k+1}\right)=\left(x_k+\Delta x,y_k+\Delta y_k\right);\left(\Delta y_k=g\left(x_k,y_k\right)\Delta x\right)$

It is possible to approximate the answer to the initial-value issue piecewise linearly by plotting this series of points and connecting them with line segments. Keep in mind that $\Delta y_k=g\left(x_k,y_k\right) \Delta x$ It is implied by$∆x$

$\frac{\Delta y_k}{\Delta x}=g\left(x_k,y_k\right)$

And from the differential equation,

$g\left(x_k,y_k\right)={\left.\frac{dy}{dx}\right|}_{\left(x_k,y_k\right)}$

Therefore, the iterative formula for determining the sequence of points required for applying Euler's technique to obtain an approximate solution to an initial-value issue uses the formula ${\left.\frac{dy}{dx}\right|}_{\left(x_i,y_2\right)}=\frac{\Delta y_k}{\Delta x}$.