Q. 54 - 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.

$\frac{dy}{dx}=\frac{1}{x},y\left(1\right)=1;\Delta x=0.25$

Step-by-Step Solution

Answer

The points are $(0,0),(0.5,0),(1,0.25),(1.5,0.875),(2,2.0625)$ and the graph is 

Step 1. Given information

Given expression $\frac{dy}{dx}=\frac{1}{x},y\left(1\right)=1;\Delta x=0.25$

Step 2: Use Euler's rule and calculate

Calculating, we get

$$\begin{gathered} \left(x_{k+1},y_{k+1}\right)=\left(x_k+\Delta x,y_k+\Delta y_k\right);\left[\Delta y_k=g\left(x_k,y_k\right)\Delta x\right] \\ \left(x_0,y_0\right)=(0,0) \\ \left(x_1,y_1\right)=\left(x_0+\Delta x,y_0+g\left(x_0,y_0\right)\Delta x\right)=(0+0.5,0+0(0.5\left)\right)=(0.5,0) \\ \left(x_2,y_2\right)=\left(x_1+\Delta x,y_1+g\left(x_1,y_1\right)\Delta x\right)=(1,0+0.5(0.5\left)\right)=(1,0.25) \\ \left(x_3,y_3\right)=\left(x_2+\Delta x,y_2+g\left(x_2,y_2\right)\Delta x\right)=(1.5,0.25+1.25(0.5\left)\right)=(1.5,0.875) \\ \left(x_4,y_4\right)=\left(x_3+\Delta x,y_3+g\left(x_3,y_3\right)\Delta x\right)=(2,0.875+2.375(0.5\left)\right)=(2,2.0625) \end{gathered}$$

Step 3: Plotting the points

Plotting, we get