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

20.3 Higher order equations 607
Higher order differential equations can be reduced to a set of first-order equations in a
similarway.
Example20.2 Express the equation
d 3 x
dt 3 −7d2 x
dt 2 +3dx dt +2x=0
as a set of first-order equations.
Solution Lettingx 1
=x,x 2
= dx
dt andx 3 = d2 x
dt 2 wefind
dx 1
dt
dx 2
dt
=x 2
=x 3
dx 3
dt
−7x 3
+3x 2
+2x 1
=0
isthe set of first-order equations representing the given differential equation.
Example20.3 (a) Expressthe coupledfirst-order equations
dy 1
dx =y 1 +y 2
dy 2
dx = 4y 1 −2y 2
as a second-order ordinary differential equation, and obtain its general solution.
(b) Express the given equations inthe form
( ) ) y
′
1
y1
=A(
y 2
y ′ 2
whereAisa2 ×2 matrix.
Solution (a) Differentiating the second of the given equations we have
d 2 y 2
dx = 4dy 1
2 dx −2dy 2
dx
and then, using the first, we find
d 2 y 2
dx =4(y 2 1 +y 2 )−2dy 2
dx
But fromthe second given equation 4y 1
= dy 2
dx +2y 2
, and therefore
d 2 y 2
dx = dy 2
2 dx +2y 2 +4y 2 −2dy 2
dx