calc
posted by Mischa .
Find m and b so that y=mx+b is a solution to the differential equation
dy/dx=1/2x+y1

I will assume you mean (1/2)x and not 1/(2x)in the differential equation
dy/dx = m, so we require that
m = x/2 + mx+ b 1
This is true if m = 1/2 and
b = 1+m = 1/2
y = x/2 + 1/2 is a solution, but not the only solution, to the differential equation. 
dy/dx=(1/2)x + y  1
I assume that is what you mean.
y = m x + b
dy/dx = m
so
m = .5 x + mx + b  1
m = (.5 + m) x + b1
well if m is to be constant, m must be .5
that means
.5 = b1
b = .5
so
y = .5 x + .5
the end 
Now check with m = .5 and b = +.5
y = .5 x + .5
dy/dx = .5
dy/dx = .5 x + (.5 x +.5)  1
= +.5  1
= .5
check
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