Solution to ((\[DifferentialD]y)/(\[DifferentialD]x))=x-(e^(x/2)), y(0)=2 is the C=4 curve of the general solution y(x)=((x^2)/2)-2(e^(x/2))+C.

plotY[C_]:= Plot[x^2/2-2*Exp[x/2]+C,{x,-4,4},PlotStyle->{Thick}]
Show[plotY[4]]
Show[plotY[8],plotY[7],plotY[6],plotY[5],plotY[4],plotY[3]]

Some sample solutions to ((\[DifferentialD]r)/(\[DifferentialD]t))=(r/2)-450,  for representative values of K in the 1-parameter family of functions r(t)=900+(Ke^(t/2)).

plotP[K_]:= Plot[900+K*Exp[t/2],{t,-4,4},PlotRange->{890,910},PlotStyle->{Thick}]

Show[plotP[-2],plotP[-1],plotP[-0],plotP[1],plotP[2],plotP[3]]