Question

ThE solution of (x-2y)dx+ydy=0 isThw solution of (x-2y)dx+ydy=0 is:

A)ln(y-x)-X/(y-x)+c
B)ln(x)+ln(y-x)=c
C)ln(y-x)-x/y-x=c
D)ln(y-x)/(x)=c
E)it cannot be solved

Answer 1

The given differential equation (x-2y)dx + ydy = 0 can be solved by separating variables and integrating. The correct solution involves natural logarithms and constants, resembling option B, although the exact form is not provided in the given options.

Step-by-step explanation:

The solution to the differential equation (x-2y)dx + ydy = 0 involves separating the variables and integrating both sides. This method assumes that you can express the equation in the form of M(x,y)dx + N(x,y)dy = 0, where M(x,y) and N(x,y) are functions of x and y, respectively. By manipulating the equation, we aim to express it in a form where we can integrate in terms of x on one side and y on the other.

Step-by-Step Solution

1. Rewrite the equation as (x-2y)dx = -ydy.
2. Divide both sides by y(x-2y) to separate the variables:
3. dx/(x-2y) = -dy/y.
4. Now, integrate both sides: ∫ dx/(x-2y) = - ∫ dy/y.
5. The integration results in ln|y-x| + C, where C is the constant of integration.
6. Exponential of both sides will give us y - x = Ce, which can also be rearranged to y = x + Ce.

Thus, the correct solution is in a form that resembles the provided options, and more specifically, it looks like the answer could be related to option B, given that it involves natural logarithms and constants. However, it should be noted that there is a slight discrepancy with the form of the options provided, and the exact answer is not listed among them.