Give an explicit solution for the given differential equation. Be sure to state the Interval on which your solution is defined. You must show all work to receive credit. dN dt 5(N – 4)
Give an explicit solution for the given differential equation. Be sure to state the Interval on which your solution is defined. You must show all work to receive credit. dN dt 5(N – 4)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Problem Statement**
Give an explicit solution for the given differential equation. Be sure to state the interval on which your solution is defined. You must show all work to receive credit.
\[ \frac{dN}{dt} = 5(N - 4) \]
**Instructions**
Solve the given differential equation and provide the explicit solution. Additionally, specify the interval over which the solution is valid. Be thorough in your work to ensure full credit.
**Equation**
\[ \frac{dN}{dt} = 5(N - 4) \]
**Steps to Solve**
1. **Separate the variables:**
\[
\frac{dN}{N - 4} = 5 \, dt
\]
2. **Integrate both sides:**
\[
\int \frac{1}{N - 4} \, dN = \int 5 \, dt
\]
3. **Solve the integrals:**
\[
\ln|N - 4| = 5t + C_1
\]
4. **Exponentiate both sides to solve for \( N \):**
\[
|N - 4| = e^{5t + C_1} = e^{5t} \cdot e^{C_1}
\]
Let \( C = e^{C_1} \), then:
\[
|N - 4| = C \cdot e^{5t}
\]
5. **Solve for \( N \):**
\[
N - 4 = \pm C \cdot e^{5t}
\]
Thus,
\[
N = 4 + Ce^{5t}
\]
where \( C \) is a constant.
**Interval of the Solution**
The solution \( N = 4 + Ce^{5t} \) is defined for all \( t \in \mathbb{R} \) since the exponential function \( e^{5t} \) is defined for all real numbers.
However, the initial conditions or additional constraints (if provided) may restrict this interval further. In the absence of such information, we assume the solution is valid for all real \( t \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa5c3d2ef-828c-4e4d-83ce-a6a69155d004%2F825b76e2-17f5-4a57-8efd-54978b5bc890%2Fmrg8u3_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement**
Give an explicit solution for the given differential equation. Be sure to state the interval on which your solution is defined. You must show all work to receive credit.
\[ \frac{dN}{dt} = 5(N - 4) \]
**Instructions**
Solve the given differential equation and provide the explicit solution. Additionally, specify the interval over which the solution is valid. Be thorough in your work to ensure full credit.
**Equation**
\[ \frac{dN}{dt} = 5(N - 4) \]
**Steps to Solve**
1. **Separate the variables:**
\[
\frac{dN}{N - 4} = 5 \, dt
\]
2. **Integrate both sides:**
\[
\int \frac{1}{N - 4} \, dN = \int 5 \, dt
\]
3. **Solve the integrals:**
\[
\ln|N - 4| = 5t + C_1
\]
4. **Exponentiate both sides to solve for \( N \):**
\[
|N - 4| = e^{5t + C_1} = e^{5t} \cdot e^{C_1}
\]
Let \( C = e^{C_1} \), then:
\[
|N - 4| = C \cdot e^{5t}
\]
5. **Solve for \( N \):**
\[
N - 4 = \pm C \cdot e^{5t}
\]
Thus,
\[
N = 4 + Ce^{5t}
\]
where \( C \) is a constant.
**Interval of the Solution**
The solution \( N = 4 + Ce^{5t} \) is defined for all \( t \in \mathbb{R} \) since the exponential function \( e^{5t} \) is defined for all real numbers.
However, the initial conditions or additional constraints (if provided) may restrict this interval further. In the absence of such information, we assume the solution is valid for all real \( t \).
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