1. You want to use Euler's method with step size 0.5 to estimatey( 1.5 ) wherey(x)is the solution to the initial value problem:

dxdyy2=2x,y(0.5)=1

a) Identify the initial values: x₀, y₀, h

b) Write the above differential equation in the following form:

dxdy=g(x,y)

Identify g(x,y)

c) Find the value of g(x₀,y₀)

d) Using Euler's method, approximate the solution of the above IVP atx₁= 1 (i.e. findy₁).

e) Find the value of g(x₁, y₁)

f) Using Euler's method, approximate the solution of the above IVP at x₂ = 1.5 (i.e. find y₂)

2.

a) Using Euler's method with step size 0.1, estimatey(0.4)wherey(x) is the solution to the initial value problem

dxdy=6+x,y(0)=1

b) Find the particular solution of the IVP

c) Find the exact solution of the IVP above at x = 0.4

d) For the above IVP, the approximation fory(0.4)from Euler's method (with step size of 0.1) is(less than/ greater than) the exact value ofy(0.4)