4 different proofs

[exzacklyright]

I'm working on these 4 proofs but I'm not sure how to prove them.

1) prove or disprove that y1(x) and y2(x) are linearly indep. functions if there exists any value of x1 of x such that the wronskian W(y1(x), y2(x) ) is not equal to zero.
I know how to take the wronskian but not sure how to show that it's indep. and not = to zero
2)Show that if y1(x) and y2(x) are linearly indep. solutions of the 2nd order differential equation: y'' + p(x)y' + q(x)y = 0 then the W(y1(x), y2(x) ) has the form k exp (-int(p(x) dx)
3) prove that if the differential eq. y'' + by' + cy = 0 has a characteristic equation with repeated roots equal to m1 then independent solutions are e^(m1x) and xe^(m1x)
4) Derive formulae for the variation of parameters solutions of a 2nd order non-homogeneous differential equation

 

[daon]

You're assuming \(\displaystyle \exists x \in X \,\, s.t.\)
\(\displaystyle \left| \begin{array}{cc} y_1(x) & y_2(x) & y_1'(x) & y_2'(x) \end{array} \right | \neq 0\)

i.e.

\(\displaystyle y_1(x)y_2'(x) - y_1'(x)y_2(x) \neq 0\)

There exist a,b such that

\(\displaystyle y_1'(x) = b \in X, y_2'(x) = a \in X\)

Hence \(\displaystyle ay_1(x) - by_2(x) \neq 0\). This means that one of \(\displaystyle a,b \neq 0\).

Then, given \(\displaystyle cy_1(x) + dy_2(x) = 0\), you need to show c=d=0.