# Particular Integral of Exponential

1. Sep 10, 2012

First-Order Linear System Transient Response

Hey there,

I am trying to solve a problem of first order equation which is

A temperature sensor has a first order response with τ = 18 seconds. The calibration curve of the sensor is presented in Figure 1. Graph the sensor response when it is exposed to the temperature profile shown in Figure 2. Present the senor voltage as a function of time. Sensor has been kept at 15 °C for a long time before the operation.

1. The problem statement, all variables and given/known data

$\tau$dy(t)/dt + y(t) = K
Figure 1 & 2 are attached

2. Relevant equations

where $\tau$ =18 and K is given by an equation 5+t/3 (deduce from figure 1)

3. The attempt at a solution

I know that first order equation’s answer is the addition of two general solutions: homogenous (natural response of the equation) and particular-integral (generated by the step function there

T(total) = T(homogenous) + T(particular-integral)
T(homogenous)= Ae^-t/5
T(particular-integral)= I am not sure how to do the integral of Ae^-t/5 ... Any Idea's

So if someone can help me with the T(particular-integral) then i would be able to find the value of A by putting the initial condition of T(0)=5V

#### Attached Files:

• ###### Figure 1 & 2.jpg
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Last edited: Sep 10, 2012
2. Sep 10, 2012

### voko

You could assume that a particular solution is at + b. Substitute this into the equation and solve for a and b.

3. Sep 10, 2012

### Vaedoris

Hint: divide both sides by tau. If you have studied ODE's, this hint should be enough.

Last edited: Sep 10, 2012