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solve the given differential equation by using an appropriate substitution. the de is homogeneous. (x − y) dx+x dy = 0

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Final answer:

The given differential equation, (x - y) dx + x dy = 0, is a homogeneous DE that can be solved through substitution and separation of variables, resulting in an equation that can be easily integrated to find the solution.

Step-by-step explanation:

We are going to solve the given homogeneous differential equation (DE): (x - y) dx + x dy = 0. A common technique is to use a substitution that will reduce this equation to a separable form. We'll take the approach of substituting v = y/x, which implies that y = vx. Taking the derivative, we get dy/dx = v + x dv/dx. We can then substitute these back into the original DE.

Substituting the values into the DE, we get:

  • (x - vx) dx + x (v + x dv/dx) dy = 0

Now, this simplifies to:

  • (1 - v) dx + x( dv + v dx ) = 0

Separating the variables and integrating both sides gives us the solution.

The substitution and separation of variables are key steps in solving this differential equation effectively.

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