Question

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.

Answer 1

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.