Question

Solve the differential equation: y' = e^y * sin(t).

Options:

A. y = ln(-sin(t))

B. y = -ln(cos(t) + C)

C. y = cos(ln(t))

D. y = e^(-cos(t))

E. None of these

Answer 1

The solution to the differential equation y' = e^y * sin(t) is found using separation of variables and integration, resulting in the solution y = -ln(cos(t) + C), which corresponds to Option B.

Step-by-step explanation:

The differential equation given is y' = e^y * sin(t). To solve this, we use separation of variables. First, move all terms containing y to one side and terms containing t to the other side:

\(rac{dy}{e^y} = sin(t) dt\)

Next, we integrate both sides of the equation:

\(\int \frac{dy}{e^y} = \int sin(t) dt\)

The integral on the left side gives -e^(-y), and the integral on the right side gives -cos(t). Including the constant of integration, we get:

\(-e^{-y} = -cos(t) + C\)

Multiplying through by -1 and taking the natural logarithm of both sides yields:

\(y = -ln(cos(t) + C)\)

The solution that matches the form of the options given in the question is Choice B:

y = -ln(cos(t) + C)