Solve the given differential equation by using an appropriate substitution. The DE is homogeneous.
$x d x + (y - 2 x) d y = 0$

Answer

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Answer

$y = x * (1 + \exp (C 1 + LambertW (- x * \exp (- C 1))) / x)$ is the final solution to the given homogeneous differential equation.

Steps

Step 1 ：Given the homogeneous differential equation $x d x + (y - 2 x) d y = 0$

Step 2 ：We use the substitution $y = v x$ . The derivative of $y$ with respect to $x$ ( $d y / d x$ ) will then be $v + x d v / d x$ .

Step 3 ：Substitute these into the differential equation and solve for $v$ .

Step 4 ：The solution to the differential equation in terms of $v$ is $v (x) = 1 + \exp (C 1 + LambertW (- x * \exp (- C 1))) / x$ .

Step 5 ：Substitute $v$ back into $y = v x$ to get the solution in terms of $y$ .

Step 6 ： $y = x * (1 + \exp (C 1 + LambertW (- x * \exp (- C 1))) / x)$

Step 7 ： $y = x * (1 + \exp (C 1 + LambertW (- x * \exp (- C 1))) / x)$ is the final solution to the given homogeneous differential equation.