Lecture 37: Homogeneous Linear Differential Equations with Constant Coefficients

38.1 Homogeneous differential equations with second-order constant coefficients

#####Definition: Secondorderhomogeneousdifferentialequationswithconstantcoefficients

description:

1. Structure: $$y^{\prime\prime}+q +py^{\prime}+qy=0$$

2. Characteristic equation: $$r^2+pr+q=0$$

Explanation

Constant coefficients and variable coefficients:
Concept:
The coefficients of an unknown function, if constant, are constant-coefficient equations; otherwise, they are variable coefficients (for example, if the coefficients include $p(x)$)
Constant coefficients:
Structure:
$y^{\prime\prime}+py^{\prime}+qy=0$
Variable coefficients:
Structure:
$y^{\prime\prime}+p(x)y^{\prime}+q(x)y=0$
Explanation:
General solution: $y=C_1y_1(x)+C_2y_2(x)$;
$y_1$ and $y_2$ are linearly independent;
$y_1/y_2$ is not a constant;
Characteristic Equations:
Concept:
The roots of characteristic equations are closely related to the solutions of differential equations;
For distinct real roots $r_1\neq r_2$:
$y=C_1e^{r_1x}+C_2e^{r_2x}$;
For a repeated real root $r_{1}=r_{2}=r$:
$y=e^{rx}(C_1+C_2x)$;
For conjugate complex roots $r_{1,2}=\alpha\pm i\beta$:
$y=e^{\alpha x}(C_1\cos\beta x+C_2\sin\beta x)$;
Note:
These formulas apply to second-order equations;
Third-order equations:
Method:
Construct one solution component for each characteristic root, accounting for multiplicity, and add the components;
Example: For $r^{3}-2r^{2}+r-2=0$:
The characteristic roots are $r_1=2,r_{2,3}=\pm i$;
Therefore, $y=C_{1}e^{2x}+C_{2}\cos x+C_{3}\sin x$;

Method