Solve by picard's method ( until to 3-th aproximation) , y'=x+cos(y) , y(0)=1

We're generating a sequence of functions with

where , so that the sequence converges to as .

:

:

:

Unless you're familiar with Fresnel integrals, you won't be able to simplify this any further.

:

My computer takes a really long time to compute , and even longer to plot it, so I've ultimately omitted from the plot. (I wonder now if by "until to 3-th aproximation" it's intended that you only go up to ...)

I've attached a plot of the approximations (dashed and colored) along with a more precisely computed numerical solution (black)

the solution is the point

step-by-step explanation:

we have

> equation a

> equation b

adds equation a and equation b

substitute the value of x in equation b to find the value of y

the solution is the point