I am a high school student. I am having some problems with the following question and can't solve it. I need help to solve this. if $y$ = $x^2\cdot \cos x$
What will be the value of: $$(x^2)d^2y/dx^2 - (4x)\cdot dy/dx + (x^2+6)y $$
I think this is likely to be more about organising your work than knowing the basic things. You want to take $u=x^2; v=\cos x$ in the product rule.
Differentiating $u$ will always give powers of $x$ or constant terms and differentiating $v$ will always give a term in sin or cos, and you will always have something derived from $u$ times something derived from $v$ - so you know your answer will be something of the form $p(x)\cos x + q(x)\sin x$.
Then you can note (using the product rule twice) that $(uv)''=u''v+2u'v'+uv''$
Then you need to make a decision whether you are going to organise terms first by the power of $x$ or alternatively by whether they have sin or cos. I have always found it helpful (if I have something a bit complicated) to take a line for each term so $$x^2y''=x^2(u''v+2u'v'+uv'')=$$
The second term goes on line 2 and the final one on line 3, with the final expressions arranged on the right hand side of the line in accordance with the scheme I have chosen. If I have done it right, I have like terms (eg all the terms in $x^2\cos x$) in columns on three lines on the right hand side and I can add them.
But if you get the calculations right, I think you will not have as many terms or as much complexity as you imagine. I've put these notes in case they help you to see that in the end.