Finding constants in a differential equation and two limit/derivative problems

[kingkong11]

Need Help, Stuck on Calc 1 Homework!

1. Find constants A, B, and C such that the function $$y = Ax^2 + Bx + C$$ satisfies the equation $$y'' + y' - 2y = x^2$$2. Find $$\lim_{x \rightarrow 0} \frac{sin(a + 2x)-2sin(a + x)+sin a}{x^2}$$3. If $$y = \frac{x}{\sqrt{a^2-1}} - \frac{2}{\sqrt{a^2-1}} \arctan\frac{\sin(x)}{a+\sqrt(a^2-1)+\cos(x)}$$ show that $$\frac{dy}{dx} = \frac{1}{a+\cos(x)}$$
I'm completely stuck with these questions; they are much harder than the ones on the exam. I'd appreciate anyone who can point me in the right direction as to where/how to start.

 

[LCKurtz]

Hello King Kong, welcome to the Forums. You are pretty much required to show your attempt before you will receive much help. Since this is your first post, I will just give you a hint on the first one. Take the derivatives, plug them in, and see what A,B, and C work.

 

[kingkong11]

Hello King Kong, welcome to the Forums. You are pretty much required to show your attempt before you will receive much help. Since this is your first post, I will just give you a hint on the first one. Take the derivatives, plug them in, and see what A,B, and C work.

Thanks for the welcome. I have attempted these problems on my own but got stuck and don't know how to continue.
For problem #1, I took the first and second derivatives of $$y = Ax^2+Bx+C$$
$$y' = 2Ax + B$$ and $$y'' = 2A$$. Then I plug them in and got:
$$2A + 2Ax + B - 2Ax^2 - 2BX - 2C = X^2$$. This is where I got stuck, what should I do next?

For problem #2, I used the addition formula for sin which is sin(x+y) = sinxcosy + cosxsiny. But the result did not help me get rid of the X^2 in the denominator.

As for problem #3, the only way I can think of is to apply the quotient and product rule, which will result in a long and tedious operation.

I appreciate any additional hint/advice.

 

[mmmboh]

For 1, on the left side of the equation there is only one term that has $$x^2$$, that is of course $$-2Ax^2$$, so from that you know that A must be -1/2, using that A you can figure out how all the other terms cancel out.

 

[kingkong11]

For 1, on the left side of the equation there is only one term that has $$x^2$$, that is of course $$-2Ax^2$$, so from that you know that A must be -1/2, using that A you can figure out how all the other terms cancel out.

I got A = -1/2 part and If I substitute -1/2 for A in the equation I get -1-x-B-2BX-2C = 0 but how do I use it to solve for other terms? Can you elaborate on that please? thanks!

 

[LCKurtz]

You want it to come out x² + 0x + 0.

 

[kingkong11]

Bump... still need help with problem 2 and 3.

 

[mmmboh]

For number 2, do you know Hopital's rule?

 

[kingkong11]

For number 2, do you know Hopital's rule?

Nope.

 

[VeeEight]

http://en.wikipedia.org/wiki/L'Hôpital's_rule

Take your numerator as a function, perhaps calling it f(x). Then call your denominator g(x) = x². What is f ''(x) and g ''(x)? Then find the limit of f ''(x) / g ''(x).