Mathematical Methods for Physicists: A concise introduction - Site Map

FIRST-ORDER DIFFERENTIAL EQUATIONS
A di€erential equation is said to be linear if each term in it is such that the
dependent variable or its derivatives occur only once, and only to the ®rst power.
Thus
d 3 y
dx 3 ‡ y dy
dx ˆ 0
is not linear, but
x 3 d 3 y
dx 3 ‡ ex sin x dy
dx ‡ y ˆ ln x
is linear. If in a linear di€erential equation there are no terms independent of y,
the dependent variable, the equation is also said to be homogeneous; this would
have been true for the last equation above if the `ln x' term on the right hand side
had been replaced by zero.
A very important property of linear homogeneous equations is that, if we know
two solutions y 1 and y 2 , we can construct others as linear combinations of them.
This is known as the principle of superposition and will be proved later when we
deal with such equations.
Sometimes di€erential equations look unfamiliar. A trivial change of variables
can reduce a seemingly impossible equation into one whose type is readily recognizable.
Many di€erential equations are very dicult to solve. There are only a relatively
small number of types of di€erential equation that can be solved in closed
form. We start with equations of ®rst order. A ®rst-order di€erential equation can
always be solved, although the solution may not always be expressible in terms of
familiar functions. A solution (or integral) of a di€erential equation is the relation
between the variables, not involving di€erential coecients, which satis®es the
di€erential equation. The solution of a di€erential equation of order n in general
involves n arbitrary constants.
First-order di€erential equations
A di€erential equation of the general form
dy
dx
…x; y† ˆf ; or g…x; y†dy ‡ f …x; y†dx ˆ 0 …2:1†
g…x; y†
is clearly a ®rst-order di€erential equation.
Separable variables
If f …x; y† and g…x; y† are reducible to P…x† and Q…y†, respectively, then we have
Q…y†dy ‡ P…x†dx ˆ 0:
…2:2†
Its solution is found at once by integrating.
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