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.
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.