Question: Use the y 1 / y 2 ratio of two functions to show whether the functions

Question

Question: Use the y1 / y2 ratio of two functions to show whether the functions are linearly independent or not. Justify your answer.

1) y1(x) = e3xcos4x , y2(x) = e3xsin4x

2) y1(x) = cos3x , y2(x) = sin3x

Please explain why this proves linear dependence or independence. Please Justify your answer thoroughly. thank you.

NOT WRONSKIAN METHOD.

for example,

y1 / y2 = cos (3x)/ sin(3x)

If we take ratio of the given functions, we will have sine and cosine functions respectively and they are not dependent upon eachother. Hence they are independent.

I DON'T UNDERSTAND WHY.

Explanation / Answer

For y1 ,y2 to be linearly dependent we must have constants a,b not both equal to 0 so that

ay1+by2=0 for all x

Now since, y1,y2 are not zero for all x hence, a and b are both non zero

Hence, y2/y1=-a/b ie a constant

So, the ratio y2/y1 must be a constant function for y1,y2 to be linearly dependent

1.)

y2/y1= tan(4x) which is not a constant function hence y1,y2 are linearly independent

2)

y2/y1=tan(3x) which is not a constant function hence y1,y2 are linearly independent

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