In Exercises 1 through 5 solve the differential equations by Heun's method. (a) Let \( h=0.2 \) and do two steps by hand calculation. Then let \( h=0.1 \) and do four steps by hand calculation. (b) Compare the exact solution \( y(0.4) \) with the two approximations in part (a). (c) Does the F.G.E. in part (a) behave as expected when \( h \) is halved? 1. \( y^{\prime}=t^{2}-y \) with \( y(0)=1, y(t)=-e^{-t}+t^{2}-2 t+2 \) 2. \( y^{\prime}=3 y+3 t \) with \( y(0)=1, y(t)=\frac{4}{3} e^{3 t}-t-\frac{1}{3} \) 3. \( y^{\prime}=-t y \) with \( y(0)=1, y(t)=e^{-t^{2} / 2} \) 4. \( y^{\prime}=e^{-2 t}-2 y \) with \( y(0)=\frac{1}{10}, y(t)=\frac{1}{10} e^{-2 t}+t e^{-2 t} \)
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