A hand-held calculator will suffice for the Problem. The problem an initial value problem and its exact solution are given. Approximate the values of x(0.2) and y(0.2)in three ways: (a) by the Euler method with two steps of size h = 0.1; (b) by the improved Euler method with a single step of size h = 0.2; and (c) by the Runge–Kutta method with a single step of size h = 0.2. Compare the approximate values with the actual values x(0.2) and y(0.2). x' =  5x - 9y, x(0) = 0, y' =  2x - y,  y(0) = -1; x(t) = 3e2t sin 3t, y(t) = e2t(sin 3t - cos 3t)

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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.