Answer
$y_1=4.2, y_2=6.216,y_3=9.6969$ and $y=3 e^{x^2+2x}$ and $y(0.6)=3 e^{(0.6)^2+2(0.6)} \approx 14.2765 $
Work Step by Step
The formula to calculate the approximations is defined as:
$y_{n+1}=y_n+f(x_n,y_n) dx$
Consider $y(1)=0$
This gives us: $x_0=0,x_1=0+0.2=0.2, x_2=0.2+0.2=0.4$
Also, $y_1=4.2, y_2=6.216,y_3=9.6969$
Now, $\int \dfrac{dy}{y}=\int 2(x+1) dx$
or, $\ln |y|=x^2+2x+c$
when $y(1)=0$, then we have $c=\ln 3$
Thus, $\ln |y|=x^2+2x+c \implies \ln |y|=x^2+2x+\ln 3$
Here, $y=e^{x^2+2x+\ln 3}$
so, $y=3 e^{x^2+2x}$
Thus, $y(0.6)=3 e^{[(0.6)^2+2(0.6)]} \approx 14.2765 $
