AB calculus question on Cengage

Question

Some of the curves corresponding to different values of C in the general solution of the differential equation are shown in the graph. Find the particular solution that passes through the point shown on the graph.

y = Ce−x/2

2y' + y = 0

Answer 1

If a point, say ( a, b ), were desired to pass thru the solution to the differential equation, then you'd solve for the constant of integration like so:

b = C * exp(-a/2)

Note: exp() and e^() are equivalent statements of the same function

b * exp(a/2) = C*exp(-a/2)*exp(a/2)

C = b * exp(a/2)

∴ y = b * exp( - ( x - a ) / 2 )

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