Consider two functions y1x xx and y2x x2 Show that the Wro

Consider two functions y_1(x) = x|x| and y_2(x) = x^2. Show that the Wronskian is identically zero, but they are independent. Explain why this example does not conflict with Theorem 2.1. Deduce that these y_1 and y_2 cannot be two solutions of any second-order linear homogeneous equation.

Solution

i)

For x >=0

y1=x^2,y2=x^2 ,hence, y1=y2

W=y1\'y2-y1y2\'=y1\'y1-y1y1\'=0

For x<0

y1=-x^2,y2=x^2,y2=-y1

W=y1\'y2-y1y2\'=-y1\'y1+y1y1\'=0

Hence Wronskian is zero for all values of x

Now lets check for linear independence

LEt, ay1+by2=0

For x>=0 we get

ax^2+bx^2=0

So, a+b=0

For x<0 we get

-ax^2+bx^2=0

So, -a+b=0

HEnce, a=b=0

HEnce the two functions are linearly independent

ii)

This does not contradict theorem 2.1 as theorem says that if the functions are lienarly dependent then

Wronskian is zero but converse is not true. That is if Wronskian is zero then functions need not be linearly dependent

Now if y1,y2 were two solutions of a second order linear homogeneous equation and since they are linearly independent then we would have that Wronskian would be non zero for all x but Wronskian here is zero so these two functions cannot be solutions