General Contents
Detailed Contents
Index
Exact Ordinary Differential Equations: Example2
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Show that the differential equation
, is exact, and find its general solution.
General Contents
Detailed Contents
Index
What is the general form of this differential equation
?.
How can we show that it is exact?
We can calculate
What is the condition on these if the differential equation is exact?
These partial derivatives are equal.
Find
Find
What is our conclusion?
The fact that they are equal means that the differential equation is exact.
What will the form of the solution be?
The solution will be a function of x and y:
U(
x
,y
).
How do we start to find this solution?
We use partial integration.
Set that up for partial integration over x.
Do the integration.
Let
;
We get
Why do we get the f(y) term?
Because we did a partial integration with respect to x.
Since
, any additional y dependence would not appear in
M(
x,y
)
.
How do we determine
f(y)
?
We can determine
from this result and compare what we get with
N(
x
,y
)
.
The two must be equal.
Take the partial derivative.
We get
Compare this to
What can we conclude about
f ’(y)
?
We see that
How can we use this result?
We can do a partial integration with respect to y.
Do it.
Combine these results to obtain the general solution.
Check this result by calculating
M(
x
,y
).
Check this result by calculating
N(
x
,y
).
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, is exact, and find its general solution.