Answer and Step-by-step explanation:
A separable differential equation is a type of differential equation that can be written in the form dy/dx = f(x)g(y), where f(x) is a function of x only and g(y) is a function of y only.
Let's go through each option and determine whether they are separable differential equations:
A. dy/dx = sin(xy)
This equation is not separable because sin(xy) is not a product of a function of x and a function of y.
B. dy/dx = √(1+xy)
This equation is not separable because √(1+xy) is not a product of a function of x and a function of y.
C. dy/dx = e^(3x+2y)
This equation is not separable because e^(3x+2y) is not a product of a function of x and a function of y.
D. y ln(x) dy/dx = (y+1/x)^2
This equation is not in the form dy/dx = f(x)g(y), but we can rearrange it to be in that form. Dividing both sides by y^2 and rearranging, we get: (y+1/x)^2/y^2 = ln(x). Now, we have a product of a function of x (ln(x)) and a function of y ([(y+1/x)^2]/y^2). Therefore, this equation is separable.
E. x√(1+y^2) dx = y√(1+x^2) dy
This equation is not in the form dy/dx = f(x)g(y), but we can rearrange it to be in that form. Dividing both sides by x√(1+y^2) and rearranging, we get: (1+x^2)^(1/2)/(1+y^2)^(1/2) = y/x. Now, we have a product of a function of x (y/x) and a function of y ([(1+x^2)^(1/2)]/[(1+y^2)^(1/2)]). Therefore, this equation is separable.
F. dy/dx = x+y
This equation is not separable because x+y is not a product of a function of x and a function of y.
Therefore, the separable differential equations among the given options are D. y ln(x) dy/dx = (y+1/x)^2 and E. x√(1+y^2) dx = y√(1+x^2) dy.