Formula
1. Euler Rule
`y_1=y_0+hf(x_0,y_0,z_0)`
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Examples
1. Find y(0.1) for `y''=1+2xy-x^2z`, `x_0=0, y_0=1, z_0=0`, with step length 0.1 using Euler method (2nd order derivative)
Solution:
Given `y^('')=1+2xy-x^2z, y(0)=1, y'(0)=0, h=0.1, y(0.1)=?`
put `(dy)/(dx)=z` and differentiate w.r.t. x, we obtain `(d^2y)/(dx^2)=(dz)/(dx)`
We have system of equations
`(dy)/(dx)=z=f(x,y,z)`
`(dz)/(dx)=1+2xy-x^2z=g(x,y,z)`
Euler method for second order differential equation
`y_1=y_0+hf(x_0,y_0,z_0)=1+(0.1)*f(0,1,0)=1+(0.1)*(0)=1+(0)=1`
`:.y(0.1)=1`
This material is intended as a summary. Use your textbook for detail explanation.
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