Question

Solve ,dy/dx=y/x ignoring lost solutions, if any group of answer choices

Answer 1

The differential equation dy/dx = y/x is solved using separation of variables, resulting in the general solution y = Cx, which represents a family of straight lines through the origin with varying slopes.

Step-by-step explanation:

To solve the differential equation dy/dx = y/x, we can use the method of separation of variables. Rearrange the equation to separate the variables y and x:

1. Divide both sides by y to get 1/y dy.
2. Multiply both sides by dx to get dx/x.
3. The equation now looks like this: 1/y dy = dx/x.
4. Integrate both sides: the left side with respect to y, and the right side with respect to x.
5. You get ln|y| = ln|x| + C, where C is the integration constant.
6. Exponentiate both sides to get rid of the natural logarithm: y = Cx.

Thus, the solution is y = Cx, where C is an arbitrary constant. This represents a family of straight lines passing through the origin, each with a different slope determined by the value of C.