The question is as follows

Let $f_1$ & $f_2$ be two solutions to the following second order homogeneous linear differential equation


a)Show that $f_1$ & $f_2$ has a common zero at a point $x_0$ of the interval $a \leq x \leq b$ then $f_1$ & $f_2$ are linearly dependent on $a \leq x \leq b$.

My work is as follows

We know that they can be written as a linear combination of solution!



where $c_1$ & $c_2$ are not both zeroes


$f_1(x_0)=0, f_2(x_0)=0$ on interval $a \leq x \leq b$

Then $f(x)=0$


For point $x_0$ on interval $a \leq x \leq b$

We have the theorem where if a system of two homogeneous linear algebraic equations has a non-trivial solution if the determinant of the system is zero.

Non-trivial means to mean me that $c_1$ & $c_2$ are not both zeroes.

Checking the determinant of the homogeneity of the linear equation!



$$W[f_1(x),f_2(x)]=\begin{bmatrix}f_1(x_0) & f_2(x_0) \\f_1'(x_0) & f_2'(x_0) \end{bmatrix}$$

$$W[f_1(x),f_2(x)]=0 $$ on interval $a \leq x \leq b$

If the system is linearly independent then it will be a contradiction.

2) Show that $f_1$ & $f_2$ have relative maxima at common point $x_0$ of interval $a \leq x \leq b$ then $f_1$ & $f_2$ are linearly dependent!

Relative maxima occurs at $x_0$ when $f_1'(x_0)$ is undefined or zeroes.



Suppose that $f_1(x_0)=k_1$ and $f_2(x_0)=k_2$

$f_1'(x_0)=0$ & $f_2'(x_0)=0$


One condition for the Wronskian definition to be zeroes that is when $f_1'(x_0)=0$ & $f_2'(x_0)=0$ if and only if $x_0$ is a relative extremum for the two function! Therefore, it is linearly dependent since wronskian is again 0.

Can someone please provide a better way of doing this?


In both cases, you have $W[f_1,f_2](x_0)=0$, so let's assume this and show that $W[f_1,f_2](x)=0$ everywhere. Let us also suppose $a_0(x)$ is nonzero on $[a,b]$. Your two solutions $f_1$ and $f_2$ satisfy \begin{align} a_0 f_2''+a_1 f_2'+a_2 f_2&=0,\\ a_0 f_1''+a_1 f_1'+a_2 f_1&=0. \end{align} Multiply the first equation by $f_1$, the second equation by $f_2$, and subtract. If $W:=f_1f_2'-f_2f_1'$, then the difference of the equations can be rewritten as a first order ODE for $W$: $$ a_0 W'+a_1W=0. $$ Observe that $W(x)=0$ is a solution to this equation on $[a,b]$ which satisfies $W(x_0)=0$. Since $a_0\neq 0$, this solution is unique, and we conclude $W(x)=0$ on $[a,b]$, implying $f_1$ and $f_2$ are linearly dependent.

  • $\begingroup$ How about the the second part of the question? $\endgroup$
    – Crazy
    Jul 10 '17 at 1:48
  • 1
    $\begingroup$ @Crazy If $f_1$ and $f_2$ have relative maximas at $x_0$, then $f_1'(x_0)=f_2'(x_0)=0$, so $W[f_1,f_2](x_0)=0$ as in the first part. $\endgroup$
    – user254433
    Jul 10 '17 at 1:49
  • $\begingroup$ Same logic to tackle the question? $\endgroup$
    – Crazy
    Jul 10 '17 at 1:50
  • $\begingroup$ Yes, the same in both parts. $\endgroup$
    – user254433
    Jul 10 '17 at 1:53

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.