Let and be two differentiable functions. The **
Wronskian** , associated to and , is the function

For a discussion on the motivation behind the Wronskian, click
HERE.

We have the following important properties:

*
*

**(1)**- If and are two solutions of the equation
*y*'' +*p*(*x*)*y*' +*q*(*x*)*y*= 0, then **(2)**- If and are two solutions of the equation
*y*'' +*p*(*x*)*y*' +*q*(*x*)*y*= 0, thenIn this case, we say that and are linearly independent.

**(3)**- If and are two linearly independent solutions
of the equation
*y*'' +*p*(*x*)*y*' +*q*(*x*)*y*= 0, then any solution*y*is given byfor some constant and . In this case, the set is called the

**fundamental set**of solutions.

**Example:** Let be the solution to the IVP

and be the solution to the IVP

Find the Wronskian of . Deduce the general solution to

**Solution:** Let us write . We know from the properties that

Let us evaluate *W*(0). We have

Therefore, we have

Since , we deduce that is a fundamental set of solutions. Therefore, the general solution is given by

,

where are arbitrary constants.

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