Q. 14 - Calculus

Question

Given an initial-value problem, we can apply Euler’s method to generate a sequence of points $(x_0, y_0), (x_1,y_1), \left(x_{2,}{y}_2\right)$, and so on. How are these coordinate points related to the solution of the initial-value problem?

Step-by-Step Solution

Answer

Ans:  This sequence of points is plotted and connected by line segments to produce a piecewise linear approximation to the solution of the initial - value problem. The approximate solution will be closer to the exact solution if a smaller value of $△x$ is chosen (say $△x=0.25$) and a large number of points are computed in the sequence.

Step 1. Given information.

given,

        $(x_0, y_0), (x_1,y_1), \left(x_{2,}{y}_2\right)$ 

Step 2. Solution:

 The method of tracing through a slope field is the graphical approach for finding an approximate solution to a differential equation. This approach is converted into an algebraic approach with the technique known as Euler's method. In this approach initial conditions are used to construct a sequence of points going forward from the given initial condition that approximate values of the solution of the given differential equation. In this approach, the initial condition is the starting point. The initial-value problem defined by $\frac{dy}{dx}=g(x,y),y\left(x_0\right)=y_0\text{ and }\mathrm{\Delta }x>0$ a sequence of points $(x_k, y_k)$ is generated by the iterative formula  $\left(x_{k+1},y_{k+1}\right)=\left(x_k+\mathrm{\Delta }x,y_k+\mathrm{\Delta }{y}_k\right);\left(\mathrm{\Delta }{y}_k=g\left(x_k,y_k\right)\mathrm{\Delta }x\right)$ 

This sequence of points is plotted and connected by line segments to produce a piecewise linear approximation to the solution of the initial - value problem. The approximate solution will be closer to the exact solution if a smaller value of $△x$ is chosen (say $△x=0.25$) and a large number of points are computed in the sequence.