First Order Linear Equations

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First Order Linear Equations

• Integrating Factors

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You could solve for y using separation of
variables to get
But what about this one?
Wait! We cant separate them
There is a way that starts with remembering the
Product Rule for Derivatives
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A first order linear differential equation is of
the form
where P and Q are continuous functions. Notice
that the equation is not separable. We are going
to have to find another way to solve for y
Lets multiply the equation by a newand
continuousfunction I(x)
Why do this?
f ?? g g?? f
It allows us to label the left side of the
equation in product rule terms like this
f I(x)
f? I(x)P(x)
g y
I?(x) I(x)P(x)
Which also implies that
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g
f
Now if we can just find a general formula for
I(x), we can work the product rule backwards.
We dont need to worry about a constant of
integration until after we solve the differential
equation which is also why we dont need a when
we dump the absolute value signs. Therefore, our
integrating factor is

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So we multiply I(x) by the differential equation
in order to solve the differential equation.
Now lets try the first one in which we already
know the answer so we can test this method
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Solve for y
First, identify P(x) and Q(x)
Now, identify I(x)
So now the equation becomes
By undo-ing the product rule on the left side,
we get
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Use substitution to solve this integral.
Now lets try the one in which we couldnt
separate the variables
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Solve for y
First, identify P(x) and Q(x)
Now, identify I(x)
So now the equation becomes
By undo-ing the product rule on the left side,
we get
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