2nd order differential equations

In summary, the conversation discusses the difficulty in finding the second derivative of a function and suggests using the chain rule in order to evaluate it. The product rule and chain rule are mentioned as relevant equations for this problem. The use of the chain rule is further explained, specifically in the case of differentiating a function of z.
  • #1
trew
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Homework Statement


xn4AxR5.png


Homework Equations

The Attempt at a Solution


I managed to find dy/dx as follows:
dXyGySZ.png


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?
 

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  • #2
trew said:
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?
 
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  • #3
trew said:

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## ?
 
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1) What is a 2nd order differential equation?

A 2nd order differential equation is a mathematical equation that involves the second derivative of a function. It is commonly used to model physical systems in science and engineering.

2) How do you solve a 2nd order differential equation?

The general method for solving a 2nd order differential equation is by finding the general solution, which involves finding the roots of the characteristic equation and using them to form a solution. Initial conditions may also be used to find a particular solution.

3) What is the difference between a 1st and 2nd order differential equation?

A 1st order differential equation involves the first derivative of a function, while a 2nd order differential equation involves the second derivative. This means that a 2nd order equation requires more information to solve, as it involves two unknown constants rather than one.

4) What are some applications of 2nd order differential equations?

2nd order differential equations are used in a variety of scientific and engineering fields, such as physics, chemistry, biology, and electrical engineering. They can be used to model the motion of objects, the growth and decay of populations, and the flow of electricity, among other things.

5) Can 2nd order differential equations have multiple solutions?

Yes, 2nd order differential equations can have multiple solutions. In fact, the general solution of a 2nd order equation usually includes two arbitrary constants, which can result in an infinite number of possible solutions depending on the initial conditions given.

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