📚 Topic Summary
Differential equations are equations that relate a function to its derivatives. Solving them often involves finding the original function. Direct integration is a straightforward method for solving differential equations, especially when the equation is in the form $\frac{dy}{dx} = f(x)$. The solution is found by integrating both sides of the equation with respect to $x$, adding a constant of integration, $C$, to account for all possible solutions. This constant represents a family of solutions. The method is applicable to separable differential equations where you can isolate the variables on each side.
🧠 Part A: Vocabulary
Match the following terms with their correct definitions:
| Term |
Definition |
| 1. Differential Equation |
A. A value that remains constant. |
| 2. Integration |
B. An equation involving a function and its derivatives. |
| 3. Constant of Integration |
C. The process of finding the antiderivative. |
| 4. Antiderivative |
D. A function whose derivative is the given function. |
| 5. Separable Equation |
E. A differential equation where variables can be isolated on either side. |
✍️ Part B: Fill in the Blanks
Direct integration is useful for solving differential equations of the form $\frac{dy}{dx} = f(x)$. We find the solution by ______ both sides with respect to $x$. Don't forget to add the ______ ______ ______ , usually denoted by $C$, because the derivative of a constant is always ______ . This constant represents a family of ______ .
🤔 Part C: Critical Thinking
Why is it important to include the constant of integration, $C$, when solving differential equations using direct integration? What does it represent?