Derive the particular solution $(3)of$ the initial$-$value prob$-$

lem $(2).$

Use the Euler method $to$ compute, $by$ hand, $y_1,y_2,$ and $y_3$

for the specified initial$-$value problem using $h=0\mathrm{.}2.$

$(a)y^{\text{'}}=-y$;$,y(0)=1$

$(b)y^{\text{'}}=2xy$;$,y(0)=0$

$(c)y^{\text{'}}=3x^2{y}^2$;$,y(0)=0$

$(d)y^{\text{'}}=1+2x{y}^2$;$,y(1)=-2$

$(e)y^{\text{'}}=2x{e}^{-y}$;$,y(1)=-1$

$(f)y^{\text{'}}=x^2-y^2$;$,y(3)=5$

$(g)y^{\text{'}}=xsiny$;$,y(0)=0$

$(h)y^{\text{'}}=tan(x+y)$;$,y(1)=2$

$(i)y^{\text{'}}=5x-2\sqrt[\phantom{2}]{y}$;$,y(0)=4$

$(j)y^{\text{'}}=\sqrt[\phantom{2}]{x+y}$;$,y(0)=3$

Program and run Euler's method for the initial$-$value prob$-$

lem $y^{\text{'}}=f(x,y),$ with $y(0)=1$ and $h=0\mathrm{.}1,$ through $y_{10}.$

Print $y_1,$dots,$y_{10}$ and the exact solution $y(x_1),$dots,$y(x_{10})as$

well. $(Six$ significant figures will suffice.$)$ Evaluate $E_{10}.$ Use

the $f(x,y)$ specified below.

$(a)2x$

$(b)-6y^2$

$(c)x+y$

$(d)ysinx$

$(e)\frac{y^2+1}{2}$

$(f)4x{e}^{-y}$

$(g)1+x^2+y$

$(h)-ytanx$

$(i)e^{x+y}$