1st order differential equation

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SUMMARY

The discussion focuses on solving the first-order differential equation given by xy' = 1 + y². The user successfully rearranged the equation into separable form, leading to the integral dy/(1+y²) = dx/x. Upon integrating both sides, they derived the solution y(x) = tan(lnx + C). The conversation clarifies that to find any specific solution, one simply needs to assign a constant value to C, confirming that the user's approach is correct and complete.

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  • Understanding of first-order differential equations
  • Knowledge of separable differential equations
  • Familiarity with integration techniques, specifically arctangent integration
  • Basic algebraic manipulation skills
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  • Study the method of solving separable differential equations in depth
  • Explore the properties and applications of the arctangent function
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Firepanda
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It says find any solution y(x) of

xy' = 1 + y^2

I rearranged into seperable form to get

dy/(1+y^2) = dx/x

Integrated both sides to get

arctan y = lnx + C

then y(x) = tan(lnx + C)

Is this ok? I'm a little struck on find 'any' solution for this, not too sure how else I could have done it.

Perhaps I didn't do it correct?
 
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I think you found all the solutions. If you want to find ANY specific solution then just set C=any constant. I'm not sure why you think you did something wrong.
 

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