# Verify a solution

• Feb 9th 2010, 10:40 PM
ChingKing08
Verify a solution
Verify that $\displaystyle xy=lny +c$ is a solution of the differential equation $\displaystyle dy/dx=(y^2)/(1-xy)$ for every value of the constant c.

I can't figure this one out...I don't even know where to start. Some help would be much appreciated. Thanks!
• Feb 9th 2010, 10:49 PM
Black
Use implicit differentiation on $\displaystyle xy=\text{ln}y+c$.
• Feb 9th 2010, 10:51 PM
dedust
Quote:

Originally Posted by ChingKing08
Verify that $\displaystyle xy=lny +c$ is a solution of the differential equation $\displaystyle dy/dx=(y^2)/(1-xy)$ for every value of the constant c.

I can't figure this one out...I don't even know where to start. Some help would be much appreciated. Thanks!

we need to show that $\displaystyle xy=lny +c$ satisfy DE $\displaystyle dy/dx=(y^2)/(1-xy)$

start from $\displaystyle xy=lny +c$, differentiate it

$\displaystyle y + xy' = \frac{y'}{y}$

$\displaystyle y(y + xy') = y'$

$\displaystyle y^2 + xyy' = y'$

$\displaystyle y^2 = y'(1 - xy)$

$\displaystyle y' = \frac{y^2}{1-xy}$

so the equation $\displaystyle xy=\ln y +c$ satisfy the differential equation $\displaystyle dy/dx=(y^2)/(1-xy)$
The text passage contains a mathematical question in the form of a verification problem for a differential equation. The solution to the problem is provided in the text.
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# Verify a solution

• Feb 9th 2010, 10:40 PM
ChingKing08
Verify a solution
Verify that $\displaystyle xy=lny +c$ is a solution of the differential equation $\displaystyle dy/dx=(y^2)/(1-xy)$ for every value of the constant c.

I can't figure this one out...I don't even know where to start. Some help would be much appreciated. Thanks!
• Feb 9th 2010, 10:49 PM
Black
Use implicit differentiation on $\displaystyle xy=\text{ln}y+c$.
• Feb 9th 2010, 10:51 PM
dedust
Quote:

Originally Posted by ChingKing08
Verify that $\displaystyle xy=lny +c$ is a solution of the differential equation $\displaystyle dy/dx=(y^2)/(1-xy)$ for every value of the constant c.

I can't figure this one out...I don't even know where to start. Some help would be much appreciated. Thanks!

we need to show that $\displaystyle xy=lny +c$ satisfy DE $\displaystyle dy/dx=(y^2)/(1-xy)$

start from $\displaystyle xy=lny +c$, differentiate it

$\displaystyle y + xy' = \frac{y'}{y}$

$\displaystyle y(y + xy') = y'$

$\displaystyle y^2 + xyy' = y'$

$\displaystyle y^2 = y'(1 - xy)$

$\displaystyle y' = \frac{y^2}{1-xy}$

so the equation $\displaystyle xy=\ln y +c$ satisfy the differential equation $\displaystyle dy/dx=(y^2)/(1-xy)$
user
The text passage contains a mathematical question in the form of a verification problem for a differential equation. The solution to the problem is provided in the text.

TYPE: SINGLE_QUESTION_WITH_ANSWER