# 2nd-Order Linear Differential Equations (2ndLDE)

2nd-Order Linear Differential Equations – 2ndLDE

A differential equation (DE) is an equation involving a function f(x) and its derivatives (x).

(Phương trình Vi Phân tuyến tính cấp 2 – Vietnamese. Phương trình)

Phương trình vi phân là 1 phương trình chứa biến độc lập x, hàm cần tìm là y = f (x) và các phương trình chứa biến x  như P(x) Q(x) và các cấp đạo hàm của nó.

Let’s forget what you have learned about Math during university 🙂

In Mechanical/Electrical engineering and Rotating equipment, the 2ndLDE is the most application in calculation of vibration, oscillation and circuit. Through this chapter, you will have the first sight of real mechanical problems evaluation.

There’re two basic facts (Theorem – định lý) enable us to solve homogeneous linear equations.

The other fact we need is given by the following theorem, which is proved in more
advanced courses. It says that the general solution is a linear combination of two linearly independent solutions y1 and y2. This means that neither nor is a constant by multiple of the other. y1(x)/y2(x) # constant

Why we have to use linearly independent solutions concept???

Proof:

For instance, the functions y1(x) = x and y2(x) = 5x are linearly dependent ( 2 nghiệm phụ thuộc tuyến tính) but y1(x) = x and y2(x) = 5x^2 are linearly independent. ( 2 nghiệm độc lập tuyến tính). Hence C1 & C1 are no more the constants but the Arbitary constants (hằng số tùy biến) —> Nghiệm tổng quát (general solution).

In general, it is not easy to discover particular solutions to a second-order linear equation. But it is always possible to do so if the coefficient functions P ,Q and  R are constant functions, that is, if the differential equation has the form:

“Sử dụng 2 nghiệm độc lập tuyến tính y1, y2 có dạng hệ số mũ e^rx để cho ra hàm số phụ, r sẽ được giải trong hàm số phụ, có r là có y1, y2 “

Assumed that:

For instance, an example: