Which of the following is the solution to the differential equation dy/dx = ey * x with initial condition y(0) = ln(4)?

To solve the differential equation dy/dx = e^y * x with the initial condition y(0) = ln(4), we can use separation of variables.

We start by rearranging the equation:

dy/e^y = x dx

Next, we integrate both sides. The left side can be integrated using the substitution u = e^y, leading to:

-e^-y = (1/2)x² + C

Now, we apply the initial condition to determine the constant C. By plugging in y(0) = ln(4), we find:

-e^-ln(4) = 0 + C

Since e^-ln(4) = 1/4, we have:

C = -1/4

Substituting this value back into our equation gives:

-e^-y = (1/2)x² – (1/4)

Multiplying through by -1 yields:

e^-y = -(1/2)x² + (1/4)

Taking the natural logarithm of both sides, we arrive at:

y = -ln( -(1/2)x² + (1/4) )

This equation satisfies the initial condition and is thus the solution to the differential equation.