What is the difference between first and second order differential equations

The first type of equation you are going to handle are the ones like:. Which can be written for both variables y and t. Now the solution of Second Order Differential Equation starts by taking a guess which is a calculated guess.

So we guess a solution to the equation of the form. Thus, we can substitute this into our differential equation as:. This equation is generally called the Auxiliary Equation which you jump straight away. This is a quadratic equation and they are easily solvable. These are the steps you need to know. The extension to our problem could be when the Right Hand Side of our 2nd Order Differential Equation may be non-zero i. This means that y c is the solution to the problem:.

And y p to the problem,. This has a second derivative d 2 y dx 2 , so is "Second Order" or "Order 2". This has a third derivative d 3 y dx 3 which outranks the dy dx , so is "Third Order" or "Order 3". Undetermined Coefficients which only works when f x is a polynomial, exponential, sine, cosine or a linear combination of those.

Variation of Parameters which is a little messier but works on a wider range of functions. We are going to use a special property of the derivative of the exponential function :. At any point the slope derivative of e x equals the value of e x :. We have reduced the differential equation to an ordinary quadratic equation!

This is a quadratic equation , and there can be three types of answer:. When it is.