Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter11: Differential Equations
Section11.1: Solutions Of Elementary And Separable Differential Equations
Problem 16E: Find the general solution for each differential equation. Verify that each solution satisfies the...
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Question
![**Problem Statement:**
Find the general form of \( f \) if \( f'(x) = -7f(x) \).
---
**Solution:**
To solve the differential equation \( f'(x) = -7f(x) \), we can use the method of separation of variables.
1. **Separate Variables:**
Rewrite the equation as:
\[
\frac{f'(x)}{f(x)} = -7
\]
2. **Integrate Both Sides:**
Integrate with respect to \( x \):
\[
\int \frac{1}{f(x)} f'(x) \, dx = \int -7 \, dx
\]
The left side simplifies to \( \ln |f(x)| \):
\[
\ln |f(x)| = -7x + C
\]
where \( C \) is the constant of integration.
3. **Solve for \( f(x) \):**
Exponentiate both sides to solve for \( f(x) \):
\[
|f(x)| = e^{C} e^{-7x}
\]
Let \( A = e^{C} \), where \( A \) is a constant. Therefore:
\[
f(x) = Ae^{-7x}
\]
Given the properties of the exponential function, the absolute value is not needed if \( A \) can be any real number since it can absorb both the positive and negative signs.
Hence, the general form of \( f \) is:
\[
f(x) = Ae^{-7x}
\]
where \( A \) is an arbitrary constant.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F70018f71-7185-4cb2-be6c-94aac78b8d2f%2F82e1d059-6300-4a02-91dc-c524957c8288%2Fvd9wdb6_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the general form of \( f \) if \( f'(x) = -7f(x) \).
---
**Solution:**
To solve the differential equation \( f'(x) = -7f(x) \), we can use the method of separation of variables.
1. **Separate Variables:**
Rewrite the equation as:
\[
\frac{f'(x)}{f(x)} = -7
\]
2. **Integrate Both Sides:**
Integrate with respect to \( x \):
\[
\int \frac{1}{f(x)} f'(x) \, dx = \int -7 \, dx
\]
The left side simplifies to \( \ln |f(x)| \):
\[
\ln |f(x)| = -7x + C
\]
where \( C \) is the constant of integration.
3. **Solve for \( f(x) \):**
Exponentiate both sides to solve for \( f(x) \):
\[
|f(x)| = e^{C} e^{-7x}
\]
Let \( A = e^{C} \), where \( A \) is a constant. Therefore:
\[
f(x) = Ae^{-7x}
\]
Given the properties of the exponential function, the absolute value is not needed if \( A \) can be any real number since it can absorb both the positive and negative signs.
Hence, the general form of \( f \) is:
\[
f(x) = Ae^{-7x}
\]
where \( A \) is an arbitrary constant.
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