Show the relation is an implicit solution of the DiffEQ

In summary, the given differential equation 2xyy' = x^2 + y^2 can be rearranged to the relation y^2 = x^2 - cx. The attempt at a solution involved using implicit differentiation to get 2yy' = 2x - c and then multiplying it by x to match the lefthand side of the original equation. This led to the realization that 2x^2 = x^2 + x^2, which helped solve the problem.
  • #1
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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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  • #2
Multiply 2yy' = 2x-c by x so that the lefthand sides look the same and use 2x^2 = x^2+x^2.
 
  • #3
thank you very much, I figured I was missing something somewhat obvious
 

FAQ: Show the relation is an implicit solution of the DiffEQ

What is an implicit solution of a differential equation?

An implicit solution of a differential equation is an equation that relates the dependent variable and the independent variable through a function that is not explicitly defined. In other words, the relation between the variables is not directly stated, but can be inferred from the differential equation.

How do you show that a relation is an implicit solution of a differential equation?

To show that a relation is an implicit solution of a differential equation, you must substitute the relation into the differential equation and verify that it satisfies the equation for all values of the independent variable. This can be done by taking the derivative of the relation and comparing it to the given differential equation.

What is the importance of finding implicit solutions of differential equations?

Finding implicit solutions of differential equations allows us to solve for the dependent variable without explicitly determining the function that relates it to the independent variable. This is useful in cases where the function cannot be easily determined or when the equation is too complex to solve explicitly.

Can a relation be both an explicit and implicit solution of a differential equation?

Yes, a relation can be both an explicit and implicit solution of a differential equation. An explicit solution is a function that can be solved explicitly for the dependent variable, while an implicit solution is a relation that satisfies the differential equation but cannot be solved explicitly. In some cases, a relation may have both an explicit and implicit solution.

What are some common examples of implicit solutions of differential equations?

Some common examples of implicit solutions of differential equations include circles, ellipses, and other conic sections. These shapes can be described by implicit equations that relate the coordinates of points on the shape without explicitly defining the function that relates them.

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