Discussion
\(y'' + p(x)y' + q(x)y = r(x)\) is a ________ ordinary differential equation.
Answer: A
It is a second order, nonhomogeneous, and linear ordinary differential equation.
The highest derivative is \(y''\), so the order is two. The unknown \(y\) and its derivatives appear only to the first power and are multiplied only by functions of \(x\), never by each other, so the equation is linear. The right-hand side \(r(x)\) does not involve \(y\); when it is not identically zero the equation is nonhomogeneous.
A homogeneous equation would have \(r(x) = 0\), so the two 'homogeneous' options are wrong, and nothing in the equation is nonlinear. 'First order' ignores the \(y''\) term.
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