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Solve the given differential equation by using an appropriate substitution. The DE is homogeneous. dy/dx = y - xy/x?

User Rick Roy
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Final answer:

To solve the given homogeneous differential equation, dy/dx = y - xy/x, by using an appropriate substitution, we can make the substitution y = ux, where u is a new variable. After solving the resulting separable equation and simplifying, we find the solution y = (C/x)x, where C is a constant.

Step-by-step explanation:

To solve the given differential equation, dy/dx = y - xy/x, by using an appropriate substitution, we can make the substitution y = ux, where u is a new variable. Then, we can find the derivatives of y and x with respect to x and substitute them into the original equation. This will yield a new equation in terms of u and x, which is easier to solve.

After substituting and simplifying, we get a new equation: x(du/dx) + u = 0. This equation is separable, and we can solve it by separating the variables and integrating both sides.

After integrating, we find that ln|u| = -ln|x| + C, where C is the constant of integration. Solving for u gives us u = C/x. Finally, substituting y = ux back into the equation gives us the solution to the original differential equation: y = (C/x)x, where C is a constant.

User P Daddy
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