Do you know the fundamentals of 2nd Order?
Calculus
Level
pending
If p and q are functions of x and solution of a differential equation y''=ky'+ly where k and l are functions of x. Then ap+bq are also solutions of above differential equation. (a and b are constants).
Depends upon k and l
Depends upon p and q
False
True in all cases
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1 solution
y=ap+bq are solutions when p and q are linear independent.
By linear independent, I mean that p/q is not constant.
Explanation:- Suppose initial conditions, y= u for x=v and y'=m for x=n is given. Then if ap+bq satisfies the above equation,
then u=at+br where t is value of p at x=v and r is value of q at x=v. Also, m=as+bd where s and d are values of p' and q' at x=n.
Now if we have got a pair of equations, whose solution will determine the existence of a and b. If Determinant of the system will exist, then it is possible to have solution of a and b.
Determinant(in expanded form) is " t d - r s" . This determinant is known as Wronskian of the given function. It can be proved that Wronskian of a linear independent solutions is non-zero in a interval (which is contained in domain of function), so the solution of the above simultaneous equation will always exist.
But if we see for linear dependent solutions, it can be easily seen that Wronskian of the system vanishes in any interval.
By linear dependent, I mean p/q is constant. So, it is not possible to obtain unique solution if equations are linearly dependent.
Note: Proof that Wronskian is non-zero for a linear independent equation can be found here.
http://en.wikipedia.org/wiki/Wronskian