1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

0.2 Nonhomogeneous Linear Equations 25

17. d

(^2) u
dt^2
=−1, u 1 (t)=1, u 2 (t)=t.

18.^1 rdrd

(

rdudr

)

=−1, u 1 (r)=1, u 2 (r)=ln(r).

19.t^2

d^2 u

dt^2 +t

du

dt−u=1, u^1 (t)=t, u^2 (t)=

1

t.
In Exercises 20–22, use Theorem 3 to develop the formula shown for a partic-
ular solution of the differential equation.

20. d

(^2) u
dt^2
+γ^2 u=f(t), up(t)=^1
γ
∫t
0
sin

(

γ(t−z)

)

f(z)dz.

21.

du

dt+au=f(t), up(t)=

∫t

0

e−a(t−z)f(z)dz.

22.

d^2 u

dt^2 −γ

(^2) u=f(t), up(t)=^1
γ
∫t
0
sinh

(

γ(t−z)

)

f(z)dz.

23.In “Model for temperature estimation of electric couplings suffering heavy

lightning currents” [A.D. Polykriti et al.,IEE Proceedings — Generation,

Transmission and Distribution, 151 (2004): 90–94], the authors model the

temperature rise above ambient in a coupling with this initial value prob-

lem:

ρcdTdt =i^2 (t)R( 1 +αT), T( 0 )= 0.

Parameters:ρis density,cis specific heat,i(t)is the current due to a light-

ening strike,Ris the resistance of the coupling at ambient temperature,

and the factor( 1 +αT)shows how resistance increases with temperature.

Simplify the differential equation algebraically to get

dT

dt =Ki

(^2) (t)(β+T), T( 0 )= 0 ,
and identifyβandKin terms of the other parameters.
24.(Continuation) The authors model the lightning current with the func-
tioni(t)=Imax(e−λt−e−μt)/n,wherenis a factor to makeImaxthe ac-
tual maximum. Obtain graphs of this function and the simpler function
i(t)=Imaxe−λt, using these values:Imax=100 kA,n= 0 .93,λ= 2 .1,
μ=150. The unit for time is milliseconds. Graph fortfrom 0 to 2 ms,
which is the range of interest.
25.(Continuation) Solve the initial value problem using the simpler function
for current. (Don’t forget to square.) Graph the result fortfrom 0 to 2 ms,
usingβ= 0 .26 andK=13.