# Difference results when solving equation with Calculus and LAPLACE

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#### david753

##### Full Member level 1
In RC circuit, if I use Calculus to solve the differential equation,
the output is Vo=Vi*(1-exp(-t/(R*C))),

But, If I use LAPLACE to solve the differential equation, the answer seems to be different. How to explain it? Or what wrong with me?

Calculus or LAPLACE?

both answers are correct but what you are doing wrong is that you are comparing the Vo(t) with h(t) which are certainly not the same , to get the right answer try convoluting the h(t) with Vi(t)=u(t) to get the Vo(t) or try to calculate the inverse laplace of Vo(s)=Vi(s)*H(s)=(1/s)*H(s) u will get the correct answer
regards,
Safwat

Re: Calculus or LAPLACE?

Dear friends,
Could u please prove both ways that can get the same result in details?
Thanks again.

Calculus or LAPLACE?

Vo(s)=Vi(s)*H(s)
Vi(s)=P/s
H(s)=wo/(s+wo)
Vo(s)=(P*wo)(s/(s+wo))
given that inverse_laplace(s*F(s0))=integration(f(t)) from 0 to t
then vo(t)=inverse_laplace(Vo(s))=integration(f(t)) from 0 to t
where f(t)=inverse_laplace((P*w0)(1/(s+wo)))=
therefore f(t)=(P*wo)exp(-wo*t)
therefore vo(t)=integration(f(t) from (0,t)
therefore vo(t)=integration((P*wo)*exp(-wo*t)) from t=0 to t=t
therefore vo(t)=P*(1-exp(-wo*t))

note that the output you write in the question is not writen correctly as you wrote vo(t)=vi(t)*(1-exp(-wo*t))
while the right answer is vo(t)=P*(1-exp(-wo*t)) as written in the link you posted
regards,
safwat

Re: Calculus or LAPLACE?

1.
Vo(s)=Vi(s)*H(s)
Vi(s)=P/s
H(s)=wo/(s+wo)
Vo(s)=(P*wo)(s/(s+wo))
It is wrong, I supposed.
The correct equation is
Vo(s)=(P*wo)(s*(s+wo))
Is it right?

2.
My question is
What does "P" mean?
Impluse or anything else?

Re: Calculus or LAPLACE?

safwatonline said:
Vo(s)=Vi(s)*H(s)
Vi(s)=P/s
H(s)=wo/(s+wo)
Vo(s)=(P*wo)(1/(s*(s+wo)))
given that inverse_laplace((1/s)*F(s))=integration(f(t)) from 0 to t
then vo(t)=inverse_laplace(Vo(s))=integration(f(t)) from 0 to t
where f(t)=inverse_laplace((P*w0)(1/(s+wo)))=
therefore f(t)=(P*wo)exp(-wo*t)
therefore vo(t)=integration(f(t) from (0,t)
therefore vo(t)=integration((P*wo)*exp(-wo*t)) from t=0 to t=t
therefore vo(t)=P*(1-exp(-wo*t))

note that the output you write in the question is not writen correctly as you wrote vo(t)=vi(t)*(1-exp(-wo*t))
while the right answer is vo(t)=P*(1-exp(-wo*t)) as written in the link you posted
regards,
safwat

i mistyped the "1/s" to be "s" so i corrected it above, but the flow + final answer is still correct.

P is the unit step amplitude u can put it as 1 (this was the Vi(t) used in the link u posted
regards,
Safwat

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