Consider the following autonomous first-order differential equation. dy dx Find the critical points and phase portrait of the given differential equation. = = (y - 5)4

An object attached to a spring undergoes simple harmonic motion modeled by the differential equation d?y + ky dt? mass of the object, and k is the spring constant. A mass of 17 kilograms stretches the spring 0.2 meters. 0 where y(t) is the displacement of the mass (relative to equilibrium) at time t, m is the m Use this information to find the spring constant. (Use g = 9.8 meters/second?) k = The previous mass is detached from the spring and a mass of 12 kilograms is attached. This mass is displaced 0.15 meters above equilibrium (above is positive and below is negative) and then launched with an initial velocity of 0.5 meters/second. Write the equation of motion in the form y(t) = c1 cos(wt) + c2 sin(wt). Do not leave unknown constants in your equation, and round all values to exactly 3 decimal places. y(t)

Let y(t) represent your bank account balance, in dollars, after t years. Suppose you start with $50000 in the account. Each year the account earns 4% interest, and you deposit $5000 into the account. This can be modeled with the differential equation: dy dt y(0) Solve this differential equation for y(t) = y(t) = = 0.04y + 5000 50000