Let *y* be the solution to the differential equation

that passes through the point (0,0). Use Euler's Method to estimate *y*(1) with step size Δ *x* = 0.25.

Hint

There will be 4 steps, involving lines at *x* = 0, *x* = 0.25, *x* = 0.5, and *x* = 0.75

Answer

**First Step:** We start at the point (0,0) and draw a line to estimate *y*(.25). The slope of this line is

We have

**Second Step:** We start at the point (.25,0) so that *y*_{old} = 0 and draw a line to estimate *y*(.5). The slope of this line is

Then we can estimate

**Third Step:** We start at the point (0.5, 0.125) so that *y*_{old} = 0.125 and draw a line to estimate *y*(0.75). The slope of this line is

Then we estimate

**Fourth Step:** We start at the point (0.75,0.375) so that *y*_{old} = 0.375 and draw a line to estimate *y*(1). The slope of this line is

Then

Our final estimate, using Euler's method, is that

*y*(1) ≅ 0.75.