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X()t = e At X( ) +0 ∫ t e A(t−τ ) B( )dττ (8.40b)
0
We illustrate the use of this solution in finding the classical motion of an
electron in the presence of both an electric field and a magnetic flux density.

Example 8.13
Find the motion of an electron in the presence of a constant electric field and
a constant magnetic flux density that are parallel.

Solution: Let the electric field and the magnetic flux density be given by:

r
E = E ê
03
r
B = B ê
03
Newton’s equation of motion in the presence of both an electric field and a
magnetic flux density is written as:
r
dv r r r
m = qE v B+ ×( )
dt
r
where v is the velocity of the electron, and m and q are its mass and charge,
respectively. Writing this equation in component form, it reduces to the fol-
lowing matrix equation:

 v   0 1 0  v  0  
d  1  = α  −   1  + β  
0
dt   v 2    1 0 0    v 2    
 v   0 0 0  v   1
3
3
qB qE
where α = 0 and β = 0 .
m m
This equation can be put in the above standard form for an inhomogeneous
first-order equation if we make the following identifications:

 0 1  0   0
 
A = α  −1 0 0  and B = β 0
   
 0 0  0   1

First, we note that the matrix A is block diagonalizable; that is, all off-diag-
onal elements with 3 as either the row or column index are zero, and therefore

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