082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

19.4 First-order linear equations: use of an integrating factor 553
Then fromthe first key pointon page 552 with µ =xandQ = 1 wehave
∫
xy = xdx= x2
2 +C
and finallyy = x 2 +C x
isthe required general solution.
Engineeringapplication19.3
RLcircuitwithrampinput
Recall from Section 2.4.7 that a standard ramp function has value 0 fort < 0 and a
valuect fort 0. This is shown in Figure 2.46. A voltage ramp signal with a value
c = 1 is applied to an RL circuit. The arrangement is shown in Figure 19.4. The
differential equation governing the currentflow,i(t), inthiscircuitisgiven by
iR+L di =t for t0 i(0)=0
dt
ShowthatthisequationcanbewrittenintheformofEquation(19.4).Henceusethe
integrating factor method tofindi(t).
i
y (t)
R
0
t
L
Figure19.4
A ramp input signal applied to aseriesRL
circuit.
Solution
This equation can bewritten as
di
dt + R L i = t for t0
L
which is a first-order linear equation.
Noteinthiscasethattheindependentvariableist andthedependentvariableisi.
Instandard form, wehave
di
+P(t)i =Q(t)
dt
whereP(t) = R L andQ(t) = t . The integrating factor, µ, isgiven by
L
µ = e ∫ P(t)dt
= e ∫ (R/L)dt
= e Rt/L ➔