Differential Equation for y = emx:

The given curve is y = emx, where e is the base of the natural logarithm and m is a constant. We need to find the differential equation that represents this curve.

To find the differential equation, we need to find the derivative of y with respect to x, which represents the rate of change of y with respect to x.

Derivative of y with respect to x:

Using the chain rule, we can find the derivative of y = emx with respect to x.

dy/dx = d(emx)/dx

To differentiate emx with respect to x, we can use the property of the exponential function:

d(emx)/dx = m * emx

Therefore, the derivative of y = emx with respect to x is dy/dx = m * emx.

Differential Equation:

Now, we have the derivative of y with respect to x as dy/dx = m * emx. To represent this as a differential equation, we need to find an expression that relates dy/dx, y, and x.

The given options are:
a) (dy/dx) = (y/x) * logx
b) (dy/dx) = (x/y) * logy
c) (dy/dx) = (y/x) * logy
d) (dy/dx) = (x/y) * logx

Analyzing the options:

Option a) (dy/dx) = (y/x) * logx:
This option does not match the derivative we found earlier, as it includes logx instead of logy.

Option b) (dy/dx) = (x/y) * logy:
This option also does not match the derivative we found earlier, as it includes x/y instead of m.

Option c) (dy/dx) = (y/x) * logy:
This option matches the derivative we found earlier, dy/dx = m * emx. Therefore, this is the correct option.

Option d) (dy/dx) = (x/y) * logx:
This option does not match the derivative we found earlier, as it includes logx instead of logy.

Therefore, the correct answer is option c) (dy/dx) = (y/x) * logy, which represents the differential equation for the curve y = emx.