1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

442 Answers to Odd-Numbered Exercises

23. Roots of characteristic equation:

m=−α±iβ, β=

√

σ^2 −α^2.

Solution of differential equation:

y(t)=e−αt

(

c 1 cos(βt)+c 2 sin(βt)

)

.

Initial conditions give:c 1 =− 0. 001 h,c 2 =(α/β)c 1.

25.v= 2 .62 m/s.

Section 0.2

1.u(t)=T+ce−at.

3.u(t)=te−at+ce−at.

5.u(t)=^1

2

tsin(t)+c 1 cos(t)+c 2 sin(t).

7.u(t)=

1

12 e

t+^1

2 te

−t+c 1 e−t+c 2 e− 2 t.

9.u(ρ)=−

1

6 ρ

(^2) +c^1
ρ+c^2.
11.h(t)=− 320 t+c 1 +c 2 e−^0.^1 t,c 1 =h 0 +3200,c 2 =−3200.
13.v(t)=t,up(t)=te−at.
15.v 1 (x)=sin(x)−ln|sec(x)+tan(x)|,v 2 =−cos(x);
up(x)=−cos(x)ln|sec(x)+tan(x)|.
17.v 1 (t)=t^2 /2,v 2 (t)=−t;up(t)=−t^2 /2.
19.v 1 (t)=− 1 / 2 t,v 2 (t)=−t/2,up(t)=−1.
23.β= 1 /α,K=Rα/ρc.
25.T=β(exp(KImax^2 ( 1 −e−^2 λt)/ 2 λ)− 1 ).
Section 0.3

1. a.u(x)=c 2 sin(x),c 2 arbitrary;

b.u(x)= 1 −cos(x)−

1 −cos( 1 )

sin( 1 ) sin(x)(unique);

c. No solution exists.

3. a. and b.λ=±( 2 n− 1 ) 2 πa,n= 1 , 2 ,...;