📚 Topic Summary
When solving a second-order linear homogeneous differential equation of the form $ay'' + by' + cy = 0$, where $a$, $b$, and $c$ are constants, we first find the characteristic equation $ar^2 + br + c = 0$. If this quadratic equation has two distinct real roots, $r_1$ and $r_2$, then the general solution to the differential equation is given by $y(x) = c_1e^{r_1x} + c_2e^{r_2x}$, where $c_1$ and $c_2$ are arbitrary constants. These constants are determined by initial conditions, if provided. The key is recognizing when the roots are real and distinct, which leads to this specific form of the general solution.
🧠 Part A: Vocabulary
Match the following terms with their definitions:
| Term |
Definition |
| 1. Characteristic Equation |
A. A solution to a differential equation that contains arbitrary constants. |
| 2. General Solution |
B. The equation obtained by substituting $y = e^{rx}$ into the differential equation. |
| 3. Real Root |
C. A root of a polynomial equation that is a real number. |
| 4. Differential Equation |
D. An equation that relates a function with its derivatives. |
| 5. Homogeneous Equation |
E. A differential equation where the right-hand side is equal to zero. |
Match the numbers to the letters for the correct definitions.
✍️ Part B: Fill in the Blanks
To solve a second-order linear homogeneous differential equation with constant coefficients, we first find the __________ equation. If the roots of this equation are real and __________, then the general solution is given by $y(x) = c_1e^{r_1x} + c_2e^{r_2x}$, where $r_1$ and $r_2$ are the __________. The constants $c_1$ and $c_2$ are determined by the __________ conditions.
🤔 Part C: Critical Thinking
Explain, in your own words, why it is important to identify whether the roots of the characteristic equation are real and distinct when solving a second-order linear homogeneous differential equation. How does the nature of the roots affect the form of the general solution?