Help me solve this equa. please? dq/dt + q/(R*C) = e/R?

Question

I have solved this equation and my result is q=e*C(1-exp(t/RC))
However, the books gives q=e*C(1-exp(-t/RC))
What's wrong with my result???

Answer 1

... dq/dt + q / RC = e / R

Integrating factor is exp(t/RC), so
... exp(t/RC) dq/dt + exp(t/RC) q / RC = e exp(t/RC) / R

... d/dt [exp(t/RC) q] = e exp(t/RC) / R

... exp(t/RC) q = e C exp(t/RC) + X

... q = C e + X / exp(t/RC)

... q = C e + X exp(-t/RC)

This is the general solution; if the integration constant is X = -C e, then
... q = C e - C e exp(-t/RC) = C e [1 - exp(-t/RC)],

just as your book says! The negative sign in front of t/RC comes from the fact that we had to divide by exp(t/RC), which is the same as multiplying with exp(-t/RC).

Answer 2

When you differentiate e*C(1-exp(t/RC)), you get a postive e*C*exp(t/RC) that won't cancel out that other positive e*C*exp(t/RC) in your equation. Hence, it has to be e*C(1-exp(-t/RC)).