Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![**Solve the initial value problem:**
\[ x^2 \frac{dy}{dx} = \frac{4x^2 - x - 2}{(x+1)(y+1)} , \quad y(1) = 2 \]
**Begin by separating the variables. Choose the correct answer below:**
**A.**
\[ \frac{x^2 (x + 1)}{4x^2 - x - 2} dy = \frac{1}{y + 1} dx \]
**B.**
\[ (y + 1) dy = \frac{4x^2 - x - 2}{x^2 (x + 1)} dx \]
**C.**
\[ \frac{dy}{dx} = \frac{4x^2 - x - 2}{x^2 (x + 1)(y + 1)} \]
**D.**
The equation is already separated.
**The solution is** \[ \boxed{\phantom{=} } \]
*(Type an implicit solution. Type an equation using \(x\) and \(y\) as the variables.)*
**Explanation of Solutions and Steps:**
In this problem, the goal is to solve the differential equation by separating the variables.
- **Option A** suggests a separation which multiplies both sides by the reciprocal of their coefficients, aiming to isolate \(dy\) and \(dx\).
- **Option B** correctly separates the variables by rearranging the given equation into integrable parts on each side, hence isolating \(dy\) on the left and \(dx\) on the right.
- **Option C** proposes a form of the equation which does not properly separate the variables due to the mixed factors of \(y\) and \(x\) on the right side.
- **Option D** incorrectly claims that the equation is already separated as given.
The correct choice, **Option B**, separates the variables to facilitate integration on each side, here:
\[ (y + 1) dy = \frac{4x^2 - x - 2}{x^2 (x + 1)} dx \]
Further steps involve integrating both sides to find the implicit form of the solution to the given initial value problem.](https://content.bartleby.com/qna-images/question/13d2fa52-7679-4dbf-8ccf-e0669bc97f10/be8eb5a9-e73f-4acf-8799-b05b755242b7/umdct38_processed.png)
Transcribed Image Text:**Solve the initial value problem:**
\[ x^2 \frac{dy}{dx} = \frac{4x^2 - x - 2}{(x+1)(y+1)} , \quad y(1) = 2 \]
**Begin by separating the variables. Choose the correct answer below:**
**A.**
\[ \frac{x^2 (x + 1)}{4x^2 - x - 2} dy = \frac{1}{y + 1} dx \]
**B.**
\[ (y + 1) dy = \frac{4x^2 - x - 2}{x^2 (x + 1)} dx \]
**C.**
\[ \frac{dy}{dx} = \frac{4x^2 - x - 2}{x^2 (x + 1)(y + 1)} \]
**D.**
The equation is already separated.
**The solution is** \[ \boxed{\phantom{=} } \]
*(Type an implicit solution. Type an equation using \(x\) and \(y\) as the variables.)*
**Explanation of Solutions and Steps:**
In this problem, the goal is to solve the differential equation by separating the variables.
- **Option A** suggests a separation which multiplies both sides by the reciprocal of their coefficients, aiming to isolate \(dy\) and \(dx\).
- **Option B** correctly separates the variables by rearranging the given equation into integrable parts on each side, hence isolating \(dy\) on the left and \(dx\) on the right.
- **Option C** proposes a form of the equation which does not properly separate the variables due to the mixed factors of \(y\) and \(x\) on the right side.
- **Option D** incorrectly claims that the equation is already separated as given.
The correct choice, **Option B**, separates the variables to facilitate integration on each side, here:
\[ (y + 1) dy = \frac{4x^2 - x - 2}{x^2 (x + 1)} dx \]
Further steps involve integrating both sides to find the implicit form of the solution to the given initial value problem.
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