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Outline for Lecture 4
4. Separable equations and applications
The following equation is called Separable equation
One can write it as
So, this equation can be integrated on both sides. Although we probably most of the time to get the y=y(x)
form, we can get the solution in
implicit
form.
is the simple (but could be implicit) form of the IVP
e.g. 1.
Solve
The solution is y(x)=7*e^(3x^2)
What if y(0) = 4? We can get the solution of y(x)=4*e^(3x^2).
e.g. 2.
Solve
The solution is (y^35y)=4xx^2+C
It’s not possible, not practical, and not necessary to find the explicit form.
If we implement IC, y(1) = 3, then we can get C=9.
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View Full DocumentThus, the solution becomes (y^35y)=4xx^2+9.
DEF:
General solution
: Solution with an arbitrary constant
Particular solution
: Solution with without the arbitrary constant
Singular solution
: Solution that can’t be obtained by setting the arbitrary constant. This is possible for
nonlinear DE.
e.g.
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 Summer '08
 Staff

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