Show the relation is an implicit solution of the DiffEQ

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SUMMARY

The differential equation 2xyy' = x^2 + y^2 can be solved by recognizing that the relation y^2 = x^2 - cx is an implicit solution. The user initially struggled with implicit differentiation, yielding the equation 2yy' = 2x - c. A key simplification involves multiplying both sides of the equation by x, allowing for a clearer comparison of terms. This approach leads to a more straightforward path to the solution.

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  • Understanding of implicit differentiation
  • Familiarity with differential equations
  • Knowledge of algebraic manipulation techniques
  • Basic calculus concepts, particularly derivatives
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  • Explore methods for solving first-order differential equations
  • Learn about algebraic simplifications in calculus
  • Review the properties of implicit functions and their derivatives
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Students studying calculus, particularly those focusing on differential equations, as well as educators looking for examples of implicit differentiation techniques.

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Homework Statement



Differential equation: 2xyy' = x^2 + y^2
Relation: y^2 = x^2 - cx

Homework Equations





The Attempt at a Solution


Hello, I can normally solve this problems with ease; however, I am having trouble with this particular problem. I have performed the implicit differentiation to get: (2yy' = 2x - c). However, I can't seem to figure out where to go from here. I am thinking that perhaps there is some obvious simplification that I am missing. If somebody can point me in the right direction I'd greatly appreciate it. Thanks.
 
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Multiply 2yy' = 2x-c by x so that the lefthand sides look the same and use 2x^2 = x^2+x^2.
 
thank you very much, I figured I was missing something somewhat obvious
 

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