Particular Integral calculation

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To calculate the particular integral for the given differential equation, one must first identify the correct method to apply, such as the method of undetermined coefficients or variation of parameters. The equation presented involves a combination of variables that complicates separation of variables, indicating a need for alternative approaches. Clarification on the specific point of confusion is essential for further guidance in solving the equation. The user has attempted to solve the equation but requires assistance to progress. Providing additional details about the attempted solution will facilitate more targeted help.
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Homework Statement



I was wondering,how do you calculate the particular integral for a differential equation that satisfies a given equation?

Homework Equations



For example an equation like this: (dy/dx)*squareroot of x - squareroot of y= x*square root of y where the values of x and y are: (x,y)=(9.4)

The Attempt at a Solution



I tried to solve this equation using the separated variables theorem but I dn't know what to do next!
 
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Well, we can't tell you what to do next if you don't tell us where you got stuck!
 
This's my attempt to the solution(see attachment)
 

Attachments

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Thread 'Distance between a Clock's hands when the distance is increasing most rapidly'

Question: A clock's minute hand has length 4 and its hour hand has length 3. What is the distance between the tips at the moment when it is increasing most rapidly?(Putnam Exam Question) Answer: Making assumption that both the hands moves at constant angular velocities, the answer is ## \sqrt{7} .## But don't you think this assumption is somewhat doubtful and wrong?
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