Q. 56 - Calculus

Question

Exercises 53-58, use Euler's method with the given $\Delta x$ to approximate four additional points on the graph of the solution $y\left(x\right)$. Use these points to sketch a piecewise-linear approximation of the solution.

$\frac{dy}{dx}=\frac{x}{1-y},y\left(0\right)=2;\Delta x=0.1$

Step-by-Step Solution

Answer

The points are $(0,2),(0.1,2),(0.2,1.99),(0.3,1.97),(0.4,1.94)$ and graph is 

Step 1. Given information

Given expression $\frac{dy}{dx}=\frac{x}{1-y},y\left(0\right)=2;\Delta x=0.1$

Step 2: Use Euler's rule and calculate

Calculating, we get

$$\begin{gathered} \left(x_{k+1},y_{k+1}\right)=\left(x_k+\Delta x,y_k+\Delta y_k\right);\left[\Delta y_k=g\left(x_k,y_k\right)\Delta x\right] \\ \left(x_0,y_0\right)=(0,2) \\ \left(x_1,y_1\right)=\left(x_0+\Delta x,y_0+g\left(x_0,y_0\right)\Delta x\right)=(0+0.1,2+0(0.1\left)\right)=(0.1,2) \\ \left(x_2,y_2\right)=\left(x_1+\Delta x,y_1+g\left(x_1,y_1\right)\Delta x\right)=(0.1+0.1,2-0.1(0.1\left)\right)=(0.2,1.99) \\ \left(x_4,y_4\right)=\left(x_3+\Delta x,y_3+g\left(x_3,y_3\right)\Delta x\right)=\left(0.3+0.1,1.97-\frac{0.3}{0.97}(0.1)\right)=(0.4,1.94) \end{gathered}$$

Step 3: Plotting the points

Plotting, we get