Please Check my Work: y'' + y = 0, y(0) = 0, y(n/2) = 2

shivers20

Junior Member
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Mar 3, 2006
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1. Solve the following boundary value problem : y" +y=0 , y(0)=0 , y(n/2)=2

Solution: General solution to y'' + y = 0 can be found using standard
diff eq techniques:

y(x) = C1*Cos(x) + C2*Sin(x)

Applying boundary values {y(0) = 0 & y(π/2) = 2}, we get:

y(x) = 2*Sin(x)

2. Use the definition of linear dependance to show that {-5, 5sin^2x, cos^x, tanx} is a linearly dependent set of functions. Do not use a Wronskian.


Solution:
You need to show that it is possible to construct at least one of the elements from a linear combination of the other three.

For example, let's try to construct
-5 = a(5sin2(x)) + b(cos2(x)) + c(tan(x))
where a, b, and c are constants.

Note that sin2(x) + cos2(x) = 1 is a constant, so let c = 0 and a = -1 and b = -5:

-1*(5sin2(x)) + -5*(cos2(x)) + 0*tan(x) = -5*(sin2(x) + cos2(x)) = -5*1 = -5

So a linear combination of two elements of the set makes a third element of the set. Thus the set is linearly dependent.
 
For the first one, check by differentiation (it works).

The boundary conditions are also met. So yes, correct.
 
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