Picard's method of succesive approximations

Question

Construct first $3$ succesive approximations $x_0,x_1,x_2$ for the following Cauchy problems:

$$x'=-x+t^2$$ $$x(0)=2$$

I have no idea how to start this... any ideas?

Answer 1

You compute successively the functions $$ x_0(t)=x_0, ~~ x_{n+1}(t)=x_0+\int_0^tf(s,x_n(s))\,ds $$ using your function $f(t,x)=-x+t^2$.

Answer 2

Rewrite your equation as

$$ x(t) = x(0) + \int_0^t[-x(u) + u^2]{\rm d}u \tag{1} $$

The ideas is to approximate this with the expression

$$ x_{n + 1}(t) = x(0) + \int_0^t[-x_n(u) + u^2]{\rm d}u \tag{2} $$

Start with

$x_0 = x(0) = 2$

Replace that in (2) and you get

$$ x_1(t) = 2 + \int_0^t [-x_0(u) + u^2]{\rm d}u = 2 - 2t + \frac{t^2}{3} \tag{3} $$

$x_1 = 2 - 2t + t^2/3$

Evaluate this in Eq. (2) again and get

$$ x_2(t) = 2 + \int_0^t[-x_1(u) + u^2] = 2 - 2t + 2t^2 + \frac{t^3}{3} - t^4 $$

I will leave the other two for you to complete