I. Introduction.
In the theory of differential equations, second-order differential equations $y'' = f(x,y'$ is a common type. This type of differential equation has a wide range of applications in many fields of science, such as physics, engineering, economics, etc. However, for some complex problems, it can be difficult to solve second-order differential equations directly. Therefore, we need to seek some reduced-order methods to convert second-order differential equations into lower-order differential equations for better understanding and solving.
2. Reduced-order second-order differential equations $y'' = f(x,y'$ type.
For reduced-order second-order differential equations $y'' = f(x,y'$ type, we can downgrade it with the following steps:
1.Introduce a new variable $z = y'$, then the original equation can be converted to $z' = f(x,z)$。
2.According to $z'= f(x,z)$ and converts it into a first-order differential equation.
3.Using the method used to solve a first-order differential equation, find the expression $z(x)$.
4.Substituting the expression of $z(x)$ back to $y = z + c$ (c is a constant) gives the solution of the original equation $y(x)$.
Below we give a concrete example to illustrate the process of downgrading.
3. Example: Solving Differential Equations $y'' - 2xy' + 2y = 0$
1.Introduce a new variable $z = y'$, then the original equation can be converted to $z' - 2xz + 2y = 0$。
2.$z'- 2xz + 2y = 0$ to $(z - 2y).' = 2xz$。
3.to $(z - 2y).'= 2xz$ to get $z - 2y = 2xz + c$ (c is constant).
4.Replace $z = y'$ substitution back $z - 2y = 2xz + c$ to get $y' - 2xy = 2xy' + c$。
5.$y' - 2xy = 2xy'+ c$ to integrate, resulting in $y = fracx +fraccx + d$ (d is constant).
IV. Conclusions. By introducing a new variable $z = y'$, we $y the second-order differential equations'' = f(x,y')$ to a first-order differential equation $z'= f(x,z)$, thus reducing the complexity of the problem. For other types of second-order differential equations, we can also try to reduce the order using a similar approach. This approach not only helps us better understand and solve second-order differential equations, but also expands our range of applications in mathematics and science.