Use the Adams Variable Step-Size Predictor-Corrector Algorithm with tolerance \( \operatorname{TOL}=10^{-4} \), hmax \( =0.25 \), and \( h \mathrm{~min}=0.025 \) to approximate the solutions to the given initial-value problems. Compare the results to the actual values.
a. \( \quad y^{\prime}=t e^{3 t}-2 y, \quad 0 \leq t \leq 1, \quad y(0)=0 ; \quad \) actual solution \( y(t)=\frac{1}{5} t e^{3 t}-\frac{1}{25} e^{3 t}+\frac{1}{25} e^{-2 t} \).
b. \( \quad y^{\prime}=1+(t-y)^{2}, \quad 2 \leq t \leq 3, \quad y(2)=1 ; \quad \) actual solution \( y(t)=t+1 /(1-t) \).
c. \( \quad y^{\prime}=1+y / t, \quad 1 \leq t \leq 2, \quad y(1)=2 ; \quad \) actual solution \( y(t)=t \ln t+2 t \).
d. \( \quad y^{\prime}=\cos 2 t+\sin 3 t, \quad 0 \leq t \leq 1, \quad y(0)=1 ; \quad \) actual solution \( y(t)=\frac{1}{2} \sin 2 t-\frac{1}{3} \cos 3 t+\frac{4}{3} \).
