Question

CALCULUS: Given the initial value problem y'=1/x and y(1)=0, use the 4 steps of Euler's method to determine an approximation of y(2). Use a step size Δx= .25 and round to 4 decimal places. i know that the answer is y(2) ≈ 0.7595and i know that y(x)= lnx but what are the steps to getting to the approx with the formula y( xo +Δx)≈y(xo)+Δx*y'(xo) where xo is the initial x value (i think) PLEASE HELP!

Answer 1

The Euler method breaks up the curve into a bunch of straight line segments, so no integration is required. I will do two steps, you can do the other two.
\[ y(x_0 + \delta x) \approx y(x_0) + \frac{1}{x_0}\cdot (\delta x)\]
Step 1:
\[x_0 = 1, y(x_0) = 0, \delta x = 0.25\]
so
\[y(1.25) \approx 0 + \frac{1}{1} \cdot 0.25 = 0.25\]
Step 2:
\[x_0 = 1.25, y(x_0) = 0.25, \delta x = 0.25\]
\[y(1.5) \approx 0.25 + \frac{1}{1.25} \cdot 0.25 = 0.4 \]

Answer 2

oops, that last number should be 0.45...

Answer 3

thank you thank you! :D