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can u plz give me the steps involved in taking the solution of differential equation.
y= c1 exp(kx)+c2 exp(-kx) into the form y= c1 sinkx+ c2 coskx.
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We are starting with a 2nd order homogeneous differential equation with constant coefficients:
And we come up with a characteristic equation with root:
Giving the general solution:
And this is a solution for all C_1 and C_2. So let's set C_1 = 1 and C_2 = 0, and then use Euler's equation.
Now we do the same with setting C_1 = 0 and C_2 = 1:
The last step is just trig identities. Now it is easy to prove that if y_1 and y_2 are solutions to a differential equation, then y_1 + y_2 and k*y_1 are solutions as well for any k. So i*y_1 - i*y_2 is a solution. Thus, we have two solutions:
Now use the Wronskian to prove that these two solutions are in fact linearly independent so long as b is not 0 (why can't b be 0?). Once you do that, we have a theorem that the most general solution to the differential equation is expressed as a linear combination of two linearly independent solutions. Therefore, the most general form is:
Since C_1 and C_2 are constants, the 2 and -2 are both "absorbed" leaving:
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
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