2nd order linear homogeneous ODE

Question

What is a 2nd order linear homogeneous ODE?

Answer 1

When we think of a second order linear ODE, we typically observe them in this form:

A(x)*y'' + B(x)*y' + C(x)*y = G(x)

where A(x), B(x), C(x), and G(x) are some functions of x. Now, if we want to have a homogeneous second order linear ODE, we want to observe an equation where the right side is not G(x), but rather, 0:

A(x)*y''+B(x)*y' + C(x)*y = 0

Sometimes we might not even have functions of x as products with y'', y' and y in our equation; they can also be constants. However, regardless of if we have constants or functions of x in our equation, we must always have a second order linear homogeneous ODE in that general form with y'', y' and y only. The equation should not have extra variables of y as products of the other differentiable terms.

Look at the equation we have:

y'' - x*sqrt(y)*y' + 2y = 0

It is true the right side of our DE is equal to 0. HOWEVER; notice our sqrt(y) term as a product with x and y' as part of our second term of the DE. Like we noticed before, a second order linear homogeneous DE SHOULD NOT have any extra terms of y as products with our differentiable terms. That means this DE is not a 2nd order linear homogeneous ODE. It is a second order DE, but it is NON linear. The answer is false in this case.