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 + xy, y(0) = 1. $
$y(0.5) \approx 1.761639264$
given an initial value problem, we were asked to use Oilers method with a specific steps size to find the value of a solution. At some point X were given differential equation y prime equals y plus X y with initial value. Y zero equals one and step size is 0.1, and you want to find why of 0.5 approximately. So since step sizes 0.1, we're starting at X zero equals zero. Well, expect five steps. We have x 00 Why zero is going to be y over x zero, which is one x one is going to be X. You're a pulsar step size just 10.1 and why one is going to be y zero plus our step size times our function y plus x y evaluated at x zero y zero just simply one. This is equal to 1.1 x two is going to be x one plus or step size 0.2. Why to is why one plus our step size times are function y plus X y Evaluated at x one y one 1.1 plus 0.1 times 1.1 This is equal to 1.2 to 1 I x three is equal to x two plus or step size, which is 20.3. Why three z people, too? Why two You're gonna start just writing out why two instead of the full value plus our step size 0.1 times that I have are function at x two y two, which is why two plus 0.2 times why, too. This is equal to 1.36752 x for his X three plus or step size. Just point for and wife for is why three plus or step size times a function evaluated at x three y three Just why three plus 30.3 times y three which is equal to 1.54 five to nine 76 approximately. And finally we have our step five x five is equal 2.4 plus 14.5 We have that. Why of 0.5 Under this approximation is about equal to why five, which is equal to y four plus our step size times the function evaluated at X for y for so wife work plus points for why, for which is equal to approximately Yeah 1.76 16 393 And if you use the store function on a calculator or have a calculator, which saves more digits, you can get a more accurate answer than this. I think this is pretty accurate.