Q10Advanced Engineering Mathematics I

Given $\frac{dy}{dx}=f(x,y)=y-x$, with initial conditions $x_0=0,y_0=2$.

Picard's iteration formula: $y_n=y_0+{\int }_0^{x}f(t,y_{n-1})dt$

First Approximation ($y_1$): $y_1=2+{\int }_0^x(y_0-t)dt=2+{\int }_0^x(2-t)dt=2+{\left[2t-\frac{t^2}{2}\right]}_0^x$ $y_1=2+2x-\frac{x^2}{2}$

Second Approximation ($y_2$): $y_2=2+{\int }_0^x(y_1-t)dt=2+{\int }_0^x(2+2t-\frac{t^2}{2}-t)dt$ $=2+{\int }_0^x(2+t-\frac{t^2}{2})dt=2+{\left[2t+\frac{t^2}{2}-\frac{t^3}{6}\right]}_0^x$ $y_2=2+2x+\frac{x^2}{2}-\frac{x^3}{6}$

Third Approximation ($y_3$): $y_3=2+{\int }_0^x(y_2-t)dt=2+{\int }_0^x(2+2t+\frac{t^2}{2}-\frac{t^3}{6}-t)dt$ $=2+{\int }_0^x(2+t+\frac{t^2}{2}-\frac{t^3}{6})dt$ $y_3=2+2x+\frac{x^2}{2}+\frac{x^3}{6}-\frac{x^4}{24}$