Used use euler's method with step size 0.1 to estimate y (0.5), where y (x) is the solution of the initial-value problem y ' = y + 4xy, y (0) = 1. (round the answer to four decimal places.)

We want to solve the Initial Value Problem y' = y + 4xy, with y (0) = 1.

To use Euler's method, define

y (i+1) = y (i) + hy' (i), for i=0,1,2, ...,

where

h = 0.1, the step size.,

x (i) = i*h

1st step.

y (0) = 1 (given) and x (0) = 0.

y (1) ≡ y (0.1) = y (0) + h*[4*x (0) * y (0) ] = 1

2nd step.

x (1) = 0.1

y (2) ≡ y (0.2) = y (1) + h*[4*x (1) * y (1) ] = 1 + 0.1 * (4*0.1*1) = 1.04

3rd step.

x (2) = 0.2

y (3) ≡ y (0.3) = y (2) + h*[4*x (2) * y (2) ] = 1.04 + 0.1 * (4*0.2*1.04) = 1.1232

4th step.

x (3) = 0.3

y (4) ≡ y (0.4) = y (3) + h*[4*x (3) * y (3) ] = 1.1232 + 0.1 * (4*0.3*1.1232) = 1.258

5th step.

x (4) = 0.4

y (5) ≡ y (0.5) = y (4) + h*[4*x (4) * y (4) ] = 1.258 + 0.1 * (4*0.4*1.258) = 1.4593

Answer: y (0.5) = 1.4593