Page 52 - Aeronautical Engineer Data Book

Fundamental dimensions and units 39
If det A = 0 but det D ≠ 0 for some i then the
i
equations are inconsistent: for example, x + y =
2, x + y = 3 has no solution.

2.8.12 Ordinary differential equations
A differential equation is a relation between a
function and its derivatives. The order of the
highest derivative appearing is the order of the
differential equation. Equations involving
only one independent variable are ordinary
differential equations, whereas those involv­
ing more than one are partial differential
equations.
If the equation involves no products of the
function with its derivatives or itself nor of
derivatives with each other, then it is linear.
Otherwise it is non-linear.
A linear differential equation of order n has
the form:
n
d y d n–1 y d y
3 3 3 3 + P n y = F
P 0 3 + P 1 n 3 + ... + P n–1
n
dx d x 1 – d x
where P i (i = 0, 1. ..., n) F may be functions of
x or constants, and P 0 ≠ 0.
First order differential equations
Form Type Method
�
�
d x y y
3 3 = f 3 3 homo- substitute u = 3 3
d y x x
geneous
d y dy
3 3 = f(x)g(y) separable ∫ 3 3 = ∫ f(x)dx + C
d x g (y )
note that roots of
g(y) = 0 are also
solutions
d y
∂ + ∂ +
g(x, y) 3 3 put 33 = f and 33
d
x ∂ x ∂y
+ f(x, y) = 0 exact = g
and solve these
∂ f ∂ g equations for +
and 3 3 = 3 3 + (x, y) = constant
∂
y ∂ x
is the solution