Calculate the first four Picard Iterates of the equation y' - y = x^2

[brit123]

Help Please!

Calculate the first four Picard Iterates of the equation y' - y = x^2 with the condition y(0) = -1

and it was given that y'(x) = x^2 +y and y(0)= -1



Need a little help with this question... Not sure what to do.

 

[AiRAVATA]

The definition of Picard iterates is

$$\phi_0=y(x_0),$$

$$\phi_{n+1}(x)=\phi_0+\int_{x_0}^x f(s,\phi_n(s))ds,$$

where $f$ is given by

[tex]y'(x)=f(x,y(x)).[/itex]<br /> <br /> In your case, $f(x,y(x))=x^2+y$, $\phi_0=y(0)=-1$, and<br /> <br /> $$\phi_1=-1+\int_0^x (s^2+\phi_0)ds=-1+\int_0^x (s^2-1)ds.$$<br /> <br /> What is $\phi_2$? and $\phi_n$?[/tex]