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**Unformatted text preview: **Differential Equations
dy d x
= (e + C ) = e x and since the right side is just y = e x + C , the differential
dx dx
equation is not satisfied except when C = 0 . However, if we instead consider y = Ce x then we
dy d
find that
= (Ce x ) = Ce x = y , so that these functions are solutions of the differential
dx dx
equation.!
Here we find that dy
in
dx
terms of either the independent or dependent variable or both. Second, the equation will usually
have an infinite number of solutions expressible by varying some arbitrary constant. This family
of solutions is called the general solution of the differential equation. To single out a unique
solution we must specify some additional information, as in the next example.
These two examples illustrate two characteristics of ODEs. First, the equation will express Example 2.3: Find the solution of dy
= y that satisfies y (0) = 10 .
dx Solution:
We use the general solution y = Ce x found in Example 2.2 (a systematic method for finding these
solutions will be discussed in the next section). Since when x = 0 we are supposed to have
y = 10 , we can substitute this information into the general solution to determine C :
10 = y (0) = Ce0 = C .
Thus, the specific solution we want is y = 10e x .!
The data consisting of a differential equation together with a numerical value of the unknown
function is called an initial value problem. Quite often, as in Example 2.3, the function value will
be the value of y at x = 0 , but a value at some other x value may be given instead. This piece of
information is referred to as the initial condition. For most problems there is a unique solution of
the differential equation that satisfies the given initial condition.
2.2 Separation of Variables
At this point the only type of ODE we can solve is one of the form y′ = f ( x) , provided we can
compute the antiderivative of f ( x) . In this section we develop a technique that can be used to
solve many differential equations. The method applies to what are called separable equations those for which the right side can be factored into a product or quotient of expressions, each of
which involves only the independent or dependent variable. 18 2 D...

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