Question

Determine whether each first-order differential equation is separable, linear, both, or neither.

dy/dx + e^xy² = x²y²
2y + e^x sinx = x³y'

Answer 1

The first equation is neither separable nor linear. The second equation is linear.

Step-by-step explanation:

The first-order differential equation dy/dx + exy² = x²y² is neither separable nor linear. To determine if an equation is separable, we need to see if it can be written in the form g(y)dy = f(x)dx. In this case, we cannot separate the variables. Linear equations have the form dy/dx + P(x)y = Q(x). However, the equation does not have this form either.

The second equation, 2y + exsin(x) = x³y', is linear. We can write it as y' - (2/x³)y = -exsin(x)/x³. This equation satisfies the form dy/dx + P(x)y = Q(x), where P(x) = -2/x³ and Q(x) = -exsin(x)/x³.