I must have missed this lecture, as with the particular solution, I am stumped, I have information but without the teaching its just garble.
Please advise on the following General Solution:
dy/dx -4y^2 = 2y
Many thanks
This DE is separable.
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To evaluate the LHS, you need to use partial fractions.
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Therefore
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Therefore and
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So .
So the DE becomes
where
where
.
Now if you have been given a boundary condition, you can solve for .
Rearranging, . Multiply by h, which is an integrating factor of sorts [this just gives us something to work with]. We have .
Notice that the LHS looks like the result of the quotient rule, so look for f and g with . Differentiating .
Looking at the denominator, set g := y. Then compare the numerators and see that f = h and \dot{f} = 2h. So we have:
where h satisfies
h is easy: . For the other, integrate from a to b to get
which will simplify if you like.