The method of variation of parameters to solve differential equations $y''+p(x)y'+Q(x)y=R(x)\ x\in I, P,Q,R$ are continuous functions for every $x\in I$ seeks a particular solution of the form $y(x)=Ay_1 +By_2$ where $y_1$ and $y_2$ are two solutions of the differental equation $y"+p(x)y'+Q(x)y=0$ and A and B are functions to be determined then which of the following are necesarily true?
(a) The wronskian of $y_1$ and $y_2$, $W(y_1,y_2)\neq 0, \forall x\in I$
(b)A and B and $Ay_1+By_2$ are two differentiable functions.
(c)A, B may not twice differentaible but $Ay_1+By_2$ is.
(d)None of these
Since the variation of parameters are used , the solutions has to be linearly independent and hence te wronskian must be non zero. so option 1 is correct.
But it is also given option 3 is also true.. I know $Ay_1+By_2$ will be twice differentiable being the solution of differentiable equation of order 2...But my question why A and B may not be twice differentiable... When we will find out second derivative of $y=Ay_1+By_2$, double derivative of A and b will also come into the picture, then how A and B may not be twice differentiable...
Thanks in advance!!