Physical systems may be described by systems of differential equations with more than one dependent variable appearing in multiple ODEs. When a chemical substance X is broken down into a product P by means of an enzyme E, the process may be described by the Michaelis-Menten reaction X + E ⇌ Y ⇌ P + E where Y is a combined substance of X and E, and the enzyme is released at the end of the process to be used again. The differential equations governing the quantities X(t) and Y(t) are given by X ′ = − k1 (E0 − Y ) X + k−1YY ′ = k1 (E0 − Y ) X − (k−1 + k2)Y where E0 is the initial concentration of enzyme E. Then (E0 − Y ) represents the amount of enzyme that is not tied up in Y, and so is available for reacting with X. Starting with initial concentrations of X(0) = 0.8 and Y(0) = 0, use Euler's method with a step size of 0.1 to determine the amount of substance X(t) still remaining at times t = 5 and t = 10. Use the parameters E0 = 0.5k1 = 0.6k−1 = 0.3k2 = 0.5