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Can someone please provide a step by step way to solve this integral and how to get the CA expression. Thanks !

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I'll use C without the subscript for simplicity.

Rearrange to dC/dt + 0.02C = 0.15.

The general solution for an equation of this form is C0(t) + C1(t) where C0(t) is the solution to the homogeneous equation

dC/dt + 0.02C = 0

That would be any multiple of exp(-0.02t).

C1(t) is the particular solution to dC/dt + 0.02C = 0.15. You can see that if C = k, a constant, then dC/dt + 0.02C = 0 + 0.02k = 0.15.

That tells you C1(t) = k = 0.15/0.02 = 7.5. It takes a little theoretical work to show this is the only possible value of C1(t), not a more complicated function.

So that tells you the most general solution to C(t) is a\*exp(-0.02t) + 7.5. What remains is to figure out what the constant a is, and you use the initial values C = 2 at t = 0 to answer that.
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So this expression is on my lecture notes but I can’t seem to figure out the integral.

Can someone please double check if the resultant expression is right and if so how I get it.

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