Q. 57 - 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.

$\frac{dy}{dx}=\frac{x}{y},y\left(0\right)=2;\Delta x=0.5$

Step-by-Step Solution

Answer

The points are $(0,2),(0.5,2),(1,2.125),(1.5,2.36),(2,2.68)$ and the graph is 

Step 1. Given information

Given expression $\frac{dy}{dx}=\frac{x}{y},y\left(0\right)=2;\Delta x=0.5$

Step 2: Use Euler's rule and calculate

Calculating, we get

$$\begin{gathered} \left(x_{k+1},y_{k+1}\right)=\left(x_k+\Delta x,y_k+\Delta y_k\right);\left[\Delta y_k=g\left(x_k,y_k\right)\Delta x\right] \\ \left(x_0,y_0\right)=(0,2) \\ \left(x_1,y_1\right)=\left(x_0+\Delta x,y_0+g\left(x_0,y_0\right)\Delta x\right)=(0+0.5,2+0(0.5\left)\right)=(0.5,2) \\ \left(x_2,y_2\right)=\left(x_1+\Delta x,y_1+g\left(x_1,y_1\right)\Delta x\right)=\left(1,2+\frac{0.5}{2}(0.5)\right)=(1,2.125) \\ \left(x_3,y_3\right)=\left(x_2+\Delta x,y_2+g\left(x_2,y_2\right)\Delta x\right)=\left(1.5,2.125+\frac{1}{2.125}(0.5)\right)=(1.5,2.36) \\ \left(x_4,y_4\right)=\left(x_3+\Delta x,y_3+g\left(x_3,y_3\right)\Delta x\right)=\left(2,2.36+\frac{1.5}{2.36}(0.5)\right)=(2,2.68) \end{gathered}$$

Step 3: Plotting the points

Plotting, we get