Second Order Homogeneous DE

二阶齐次微分方程

A linear second order homogeneous differential equation involves terms up to the second derivative of a function. For the case of constant multipliers, The equation is of the form

一阶二阶齐次微分方程包含函数的二阶导数项。对于常系数的情况，方程形式为

and can be solved by the substitution

并且可以通过变量替换来求解

The solution which fits a specific physical situation is obtained by substituting the solution into the equation and evaluating the various constants by forcing the solution to fit the physical boundary conditions of the problem at hand. Substituting gives

为特定物理情况找到解，是将解代入方程，并通过强制解满足问题的物理边界条件来确定各种常数。代入后得到

which leads to a variety of solutions, depending on the values of a and b. In physical problems, the boundary conditions determine the values of a and b, and the solution to the quadratic equation for λ reveals the nature of the solution.

这导致了多种解，具体取决于a和b的值。在物理问题中，边界条件决定了a和b的值，而对λ的二次方程的求解揭示了解的性质。

Case I: Two real roots

情况一：两个实根

For values of a and b such that

there are two real roots λ₁ and λ₂ which lead to a general solution of the form

对于某些a和b的值，使得存在两个实根λ₁和λ₂，从而得到通解的形式为

Case II: A real double root

情况二：一个实重根

If a and b are such that

then there is a double root λ =-a/2 and the unique form of the general solution is

如果 a 和 b 满足某种条件，则存在二重根 λ = -a/2，且通解的唯一形式为

Case III: Complex conjugate roots

第三种情况：共轭复根

For values of a and b such that

there are two complex conjugate roots of the form and the general solution is

对于使得根为复共轭形式的a和b的值，通解为

The solution to the homogeneous equation is important on its own for many physical applications, and is also a part of the solution of the non-homogeneous equation.