2nd order differential equations

[trew]

Homework Statement

Homework Equations

The Attempt at a Solution

I managed to find dy/dx as follows:

But I'm having difficulty finding the second derivative. I've looked at examples using the chain rule but I'm still confused.

Would someone mind shedding some light on this for me?

 

[Orodruin]

I managed to find dy/dx as follows:

This is not particular for dy/dx. It holds if you replace y by any function. In particular, what do you get if you replace y by dy/dx?

 

[Stephen Tashi]

Homework Equations

The product rule and the chain rule are relevant equations.

But I'm having difficulty finding the second derivative.

Part of applying the product rule to find $\frac{d^2 y}{dx^2} = \frac{d}{dx} ( \frac{dy}{dz} cos(x))$ requires evaluating the factor $\frac{d}{dx} \frac{dy}{dz}$. The chain rule, in words, says "the derivative with respect to x of a function of z is equal to the derivative of the function with respect to z times the derivative of z with respect to x". As @Orodruin pointed out, you can apply this rule when "a function of z" is $\frac{dy}{dz}$. What do you get when you differentiate $\frac{dy}{dz}$ with respect to $z$ ?