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Find out some solutions to
y′′ - 9 y = 0
Solution
We can find some solutions here simply through inspection. We require functions whose second derivative is 9 times the original function. One of the first functions which I can think of that comes back to it after two derivatives is an exponential function and along with proper exponents the 9 will find taken care of as well.
Therefore, it looks like the subsequent two functions are solutions.
y(t) = e3t and y(t) = e-3t
We'll leave this to you to verify that these are actually solutions.
These two functions are not the merely solutions to the differential equation though. Any of the subsequent is also solutions to the differential equation.
y (t ) = -9e3t
y (t ) = 56e-3t
y (t ) = 7e3t - 6e-3t
y (t ) = 123e3t
y (t ) = (14/9) e-3t
y (t )= -92e3t -16e-3t
Actually, if you think about it any function which is in the form
y (t ) = c e3t + c e-3t will be a solution to the differential equation.
This illustration leads us to a very significant fact that we will use in practically each problem in this section will be a solution to the differential equation.
Consider the given graph G below. Find δ( G )=_____ , λ( G )= _____ , κ( G )= _____, number of edge-disjoint AF -paths=_____ , and number of vertex-disjoint AF -paths= ______
L.H.S. =cos 12+cos 60+cos 84 =cos 12+(cos 84+cos 60) =cos 12+2.cos 72 . cos 12 =(1+2sin 18)cos 12 =(1+2.(√5 -1)/4)cos 12 =(1+.(√5 -1)/2)cos 12 =(√5 +1)/2.cos 12 R.H.S =c
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