91Ó°ÊÓ

We are given the following second-order differential equation: \(y^{\prime \prime}(t)+9y(t)=10.\)

A second-order differential equation is linear if it can be written in the form: \(a(t)y^{\prime \prime}(t) + b(t)y^{\prime}(t) + c(t)y(t) = d(t)\) where \(a(t)\), \(b(t)\), \(c(t)\), and \(d(t)\) are continuous functions of \(t\). A differential equation is nonlinear if it does not fit this form. For the given differential equation, we don't have a \(y^{\prime}(t)\) term. We can rewrite the equation in the general linear form: \(y^{\prime \prime}(t) + 0y^{\prime}(t) + 9y(t) = 10\) Comparing the given equation to the general linear form, we see that: - \(a(t)=1\), which is continuous - \(b(t)=0\), which is continuous - \(c(t)=9\), which is continuous - \(d(t)=10\), which is continuous All the functions \(a(t)\), \(b(t)\), \(c(t)\) and \(d(t)\) are continuous functions of \(t\), so the given equation fits the general linear form.

Since the given second-order differential equation fits the general linear form, we can conclude that the equation: \(y^{\prime \prime}(t)+9y(t)=10\) is a linear differential equation.