Find all the solutions of the first-order differential equations. When an initial condition is given, find the particular solution satisfying that condition.

a. \(\frac{d y}{d x}=\frac{e^{x}}{2 y}\).

b. \(\frac{d y}{d t}=y^{2}\left(1+t^{2}\right), y(0)=1\).

c. \(\frac{d y}{d x}=\frac{\sqrt{1-y^{2}}}{x}\).

d. \(x y^{\prime}=y(1-2 y), \quad y(1)=2\).

e. \(y^{\prime}-(\sin x) y=\sin x\).

f. \(x y^{\prime}-2 y=x^{2}, y(1)=1\).
g. \(\frac{d s}{d t}+2 s=s t^{2}, \quad s(0)=1\).
h. \(x^{\prime}-2 x=t e^{2 t}\).
i. \(\frac{d y}{d x}+y=\sin x, y(0)=0\).
j. \(\frac{d y}{d x}-\frac{3}{x} y=x^{3}, y(1)=4\).