Tessellate S.T.E.M.S (2019) - Physics - Category C - Set 2 - Objective Problem 1

Here are the Euler Lagrange equations:

$\large{\frac{d}{dt} \frac{\partial{L}}{\partial{\dot{x}}} = \frac{\partial{L}}{\partial{x}} \\ \frac{d}{dt} \frac{\partial{L}}{\partial{\dot{y}}} = \frac{\partial{L}}{\partial{y}}}$

Evaluating the $x$ EL equation step by step:

$\large{\frac{\partial{L}}{\partial{\dot{x}}} = m \dot{y} + b y \\ \frac{d}{dt} \frac{\partial{L}}{\partial{\dot{x}}} = m \ddot{y} + b \dot{y} \\ \frac{\partial{L}}{\partial{x}} = -b \dot{y} - ky \\ \boxed{m \ddot{y} + 2 b \dot{y} + k y = 0}}$

Evaluating the $y$ EL equation step by step:

$\large{\frac{\partial{L}}{\partial{\dot{y}}} = m \dot{x} - b x \\ \frac{d}{dt} \frac{\partial{L}}{\partial{\dot{y}}} = m \ddot{x} - b \dot{x} \\ \frac{\partial{L}}{\partial{y}} = b \dot{x} - kx \\ \boxed{m \ddot{x} - 2 b \dot{x} + k x = 0}}$

Differentiate the langrangian with respect to the time derivative of X, then differentiate the resulting expression with respect to time. Subtract from this result, the derivative of the langrangian with respect to X. Repeat this process for Y and it's derivative and the answer follows.