determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false

a) the function f(x)=3/2 +cx^-2 is a solution of the differential equation xy'+2y=3

b) the differential equation dy/dx= (f(x)g(y))/ (F(x)+G(y) is separable.

a)

y=3/2 + c/x^2
y' = -2c/x^3

xy' + 2y = x(-2c/x^3) + 2(3/2 + c/x^2)
= -2c/x^2 + 3 + 2c/x^2 = 3

b)
dy/dx = f(x)g(y)/(F(x)+G(y))
dy/g(y) (F(x)+G(y)) = f(x)dx

There's no way to separate the variables. That pesky F(x) is stuck fast to dy, or G(y) is stuck to dx.

a) To determine whether the function f(x) = 3/2 + cx^(-2) is a solution of the differential equation xy' + 2y = 3, we need to substitute the function into the differential equation and check if it satisfies it.

First, let's find the derivative of f(x): f'(x) = -2cx^(-3).

Now, substitute the function and its derivative into the differential equation:
x(3/2 + cx^(-2))' + 2(3/2 + cx^(-2)) = 3.

Expanding and simplifying:
(3/2)x + cx^(-1) - 4cx^(-3) + 3 = 3.

Simplifying further:
(3/2)x + cx^(-1) - 4cx^(-3) = 0.

From the above equation, you can observe that the left side does not reduce to zero for all values of x, which means that the given function f(x) = 3/2 + cx^(-2) is not a solution to the differential equation xy' + 2y = 3.

Example: Let's take c = 1 and substitute it into the given function: f(x) = 3/2 + x^(-2). If x = 1, the function becomes f(1) = 3/2 + 1 = 5/2. Now, let's check if it satisfies the given differential equation: 1(5/2) + 2(5/2) ≠ 3. Hence, we find that f(x) = 3/2 + x^(-2) is not a solution to the differential equation.

b) To determine whether the differential equation dy/dx = (f(x)g(y))/(F(x) + G(y)) is separable, we need to check if we can separate the variables x and y on each side of the equation.

From the given equation, we can rearrange it as follows:
(F(x) + G(y))dy = f(x)g(y)dx.

Now, if we can find functions F(x) and G(y) such that the left side only contains y variables, and the right side contains only x variables, then the differential equation is separable.

However, we cannot generally separate the variables in this equation since F(x) and G(y) may depend on both x and y variables. Therefore, the differential equation dy/dx = (f(x)g(y))/(F(x) + G(y)) is not separable.

Example: Consider the equation dy/dx = (x^2y)/(x + y^2). If we try to separate the variables, we get (x + y^2)dy = x^2ydx. However, both x and y variables appear on both sides of the equation, so we cannot fully separate the variables.