We are working on finding the particular solution for the first-order linear constant-coefficient ordinary differential equation:
RCdvP(t)dt+vP(t)=Vicos(ωt)
The new idea is to find the particular solution for the related equation
RCdvPS(t)dt+vPS(t)=Viest
because we expect that this solution is easy to get and it will help to find the particular solution for the original equation.
If you try vPS(t)=Vpest you will find that it will work; you can solve for Vp. In the space provided below write an algebraic expression for Vp in terms of Vi, s, R and C.
To find the particular solution, we substitute the assumed solution vPS(t) = Vpest into the equation RCdvPS(t)dt + vPS(t) = Viest.
Substituting vPS(t) with Vpest and its derivative with Vpes, the equation becomes:
RC(Vpes)dt + Vpest = Viest
Since e^st is a common factor, we can factor it out:
e^st(RCVp + Vp) = Viest
Now, we can divide both sides of the equation by e^st:
RCVp + Vp = Vie
Factoring out Vp on the left side of the equation:
Vp(RC + 1) = Vie
Finally, we can solve for Vp by dividing both sides by (RC + 1):
Vp = Vie / (RC + 1)
So, the algebraic expression for Vp in terms of Vi, s, R, and C is:
Vp = Vie / (RC + 1)