Verify that the indicated function y = p(x) is an explicit solution of the given first-order differential equation. (y - x)y' = y - x + 18; y = x + 6√x + 3 When y = x + 6√x + 3, y' = Thus, in terms of x, (y - x)y' = y-x + 18 = Since the left and right hand sides of the differential equation are equal when x + 6√x + 3 is substituted for y, y = x + 6√x + 3 is a solution. Proceed as in Example 6, by considering simply as a function and give its domain. (Enter your answer using interval notation.) Then by considering as a solution of the differential equation, give at least one interval I of definition. (-3, ∞0) (-6, -3] (-6, 3) (-∞, -3) O [-3, 3] D
Verify that the indicated function y = p(x) is an explicit solution of the given first-order differential equation. (y - x)y' = y - x + 18; y = x + 6√x + 3 When y = x + 6√x + 3, y' = Thus, in terms of x, (y - x)y' = y-x + 18 = Since the left and right hand sides of the differential equation are equal when x + 6√x + 3 is substituted for y, y = x + 6√x + 3 is a solution. Proceed as in Example 6, by considering simply as a function and give its domain. (Enter your answer using interval notation.) Then by considering as a solution of the differential equation, give at least one interval I of definition. (-3, ∞0) (-6, -3] (-6, 3) (-∞, -3) O [-3, 3] D
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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Question
![### Verifying Explicit Solutions to First-Order Differential Equations
**Objective:**
Verify that the indicated function \( y = \varphi(x) \) is an explicit solution of the given first-order differential equation.
Given Differential Equation:
\[
(y - x)y' = y - x + 18; \quad y = x + 6\sqrt{x} + 3
\]
**Procedure:**
1. **Substitute the given function \( y = x + 6\sqrt{x} + 3 \) into the differential equation.**
When \( y = x + 6\sqrt{x} + 3 \),
\[
y' = \_\_\_\_\_\_\_
\]
2. **Compute the derivative \( y' \).**
3. **Express the differential equation in terms of \( x \).**
Thus, in terms of \( x \),
\[
(y - x)y' = \_\_\_\_\_\_\_
\]
\[
y - x + 18 = \_\_\_\_\_\_\_
\]
4. **Verify that the left and right-hand sides of the differential equation are equal:**
Since the left and right-hand sides of the differential equation are equal when \( x + 6\sqrt{x} + 3 \) is substituted for \( y \),
\[
y = x + 6\sqrt{x} + 3
\]
is a solution.
5. **Determine the domain of the function \( \varphi \):
Proceed as in Example 6, by considering \( \varphi \) simply as a function and give its domain. (Enter your answer using interval notation.)
\[
\_\_\_\_\_\_\_
\]
6. **Identify at least one interval \( I \) of definition:**
Then by considering \( \varphi \) as a solution of the differential equation, give at least one interval \( I \) of definition.
\[
\_\_\_\_\_
\]
**Multiple Choice Question:**
Identify the correct interval from the options below:
- \(( -3, \infty )\)
- \(( -6, -3 ]\)
- \(( -6, 3 )\)
- \(( -\infty, -3 )\)
- \([](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0436118d-47b7-4fa9-abd3-dac72bbeccd1%2F9a59603e-aa9f-4cd3-a436-95a993674980%2Fnd3y98_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Verifying Explicit Solutions to First-Order Differential Equations
**Objective:**
Verify that the indicated function \( y = \varphi(x) \) is an explicit solution of the given first-order differential equation.
Given Differential Equation:
\[
(y - x)y' = y - x + 18; \quad y = x + 6\sqrt{x} + 3
\]
**Procedure:**
1. **Substitute the given function \( y = x + 6\sqrt{x} + 3 \) into the differential equation.**
When \( y = x + 6\sqrt{x} + 3 \),
\[
y' = \_\_\_\_\_\_\_
\]
2. **Compute the derivative \( y' \).**
3. **Express the differential equation in terms of \( x \).**
Thus, in terms of \( x \),
\[
(y - x)y' = \_\_\_\_\_\_\_
\]
\[
y - x + 18 = \_\_\_\_\_\_\_
\]
4. **Verify that the left and right-hand sides of the differential equation are equal:**
Since the left and right-hand sides of the differential equation are equal when \( x + 6\sqrt{x} + 3 \) is substituted for \( y \),
\[
y = x + 6\sqrt{x} + 3
\]
is a solution.
5. **Determine the domain of the function \( \varphi \):
Proceed as in Example 6, by considering \( \varphi \) simply as a function and give its domain. (Enter your answer using interval notation.)
\[
\_\_\_\_\_\_\_
\]
6. **Identify at least one interval \( I \) of definition:**
Then by considering \( \varphi \) as a solution of the differential equation, give at least one interval \( I \) of definition.
\[
\_\_\_\_\_
\]
**Multiple Choice Question:**
Identify the correct interval from the options below:
- \(( -3, \infty )\)
- \(( -6, -3 ]\)
- \(( -6, 3 )\)
- \(( -\infty, -3 )\)
- \([
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