1. For the first order ordinary differential equation, the general solution formula is: dy/dx = f(x)dydx = f(x)dx.
2. For the second-order homogeneous linear differential equation with constant coefficients, for example: y+py+qy=0, the general solution formula is: y=e? ∫p(x)dx(c+∫q(x)e∫p(x)dxdx).
How are these general solutions derived? First of all, we need to know what the solution of differential equation is. A solution is a function that can make the left and right sides of a differential equation equal. For first-order differential equations, we usually set y=f(x) as the solution of the equation, and then substitute it into the equation to find the expression of f(x). For differential equations of second order and above, we need to use more complicated skills to solve them.
For the second-order homogeneous linear differential equation with constant coefficients, we can turn it into a set of first-order differential equations, and then solve it with the general solution formula of first-order differential equations. This process requires mathematical integration and differential operations.
Application of differential equation;
Physics: In physics, differential equations are used to describe many natural phenomena. For example, Newton's second law is a typical example. It uses differential equations to describe the motion state of objects, which provides an important tool for studying the motion of objects. In addition, when studying thermodynamics, electromagnetism and other fields, differential equations are also needed to establish mathematical models and solve problems.
2. Biology: In biology, differential equations are used to describe the growth and evolution of biological populations. For example, the famous logistic equation is an example, which describes the relationship between the growth rate of biological population and resource limitation. In addition, when studying the spread of epidemic diseases and the balance of ecosystem, it is also necessary to establish a mathematical model by using differential equations to predict the future development trend.
3. Engineering: In engineering, differential equations are used to describe the behavior of many dynamic systems. For example, in control theory, signal processing, economics and other fields, it is necessary to use differential equations to establish mathematical models and optimize the performance of the system. In addition, when studying mechanical vibration, electronic circuits and other problems, it is also necessary to use differential equations to establish mathematical models to solve practical problems.