Second order linear differential equations

Many physical systems can be described by differential equations. A second order linear differential equation has the form,

$\large a \frac{d^2x}{dt^2}+ b\frac{dx}{dt}+cx = f(t).$

Here $a$, $b$, and $c$ are constants and $f$ is a function of $t$. A second order differential equation like this can always be written as two coupled first order differential equations. Introducing $v=\frac{dx}{dt}$, this equation can be written,

$\large v=\frac{dx}{dt},$

$\large a \frac{dv}{dt}+ bv+cx = f(t).$

Second order differential equations are common in physics but can be difficult to solve. For the questions in this section, you will use a 2nd order differential equation solver to find the solutions. You will not have to solve the differential equations on your own. Your task will be to read a question that can be solved using differential solver and plugging the correct parameters into the solver to get the solution.