Standard +0.3 This is a standard second-order linear ODE with constant coefficients requiring the complementary function (repeated root λ=-2) and particular integral (exponential form). The second part involves straightforward limit analysis. While it's a Further Maths topic, the techniques are routine and well-practiced, making it slightly easier than average overall.
Find the general solution of the differential equation
$$\frac{d^2y}{dt^2} + 4\frac{dy}{dt} + 4y = e^{-\alpha t},$$
where \(\alpha\) is a constant and \(\alpha \neq 2\). [7]
Show that if \(\alpha < 2\) then, whatever the initial conditions, \(ye^{\alpha t} \to \frac{1}{(2-\alpha)^2}\) as \(t \to \infty\). [2]
Find the general solution of the differential equation
$$\frac{d^2y}{dt^2} + 4\frac{dy}{dt} + 4y = e^{-\alpha t},$$
where $\alpha$ is a constant and $\alpha \neq 2$. [7]
Show that if $\alpha < 2$ then, whatever the initial conditions, $ye^{\alpha t} \to \frac{1}{(2-\alpha)^2}$ as $t \to \infty$. [2]
\hfill \mbox{\textit{CAIE FP1 2003 Q7 [9]}}