2 1. Solve the following differential equation: y′′(x) + 6y′(x) + 25y(x) = 0. 3 2. Consider a linear differential equation ay′′ + by′ + cy = f(x) with constant coefficients. Let y1(x), y2(x) be...

Hi there is 7 math question which can be written by hand I don't mind but I need it in 1 hour and half can you help?


2 1. Solve the following differential equation: y′′(x) + 6y′(x) + 25y(x) = 0. 3 2. Consider a linear differential equation ay′′ + by′ + cy = f(x) with constant coefficients. Let y1(x), y2(x) be solutions to the homogeneous equation ay′′ + by′ + cy = 0 and let yp(x) be a particular solution Prove that C1y1 + C2y2 + yp is a solution for any choice of constants C1, C2. 4 3. A swimming pool whose volume is 10,000 gal contains water that is 0.01% chlorine. Starting at t = 0, city water containing 0.001% chlorine is pumped into the pool at a rate of 5 gal/min. The pool water flows out at the same rate. What is the percentage of chlorine in the pool after 1 h? When will the pool water be 0.002% chlorine? 5 4. There are infinitely many different solutions to y′′ − y = 0 with conditions y(0) = 0 and y′(0) = 1. This is TRUE / FALSE. Reason: 5. Solve the following differential equation: 6 y′′′ + y′′ + 4y′ + 4y = 0. 7 6. Find the general solution of the differential equation: y′′′ − y′′ + y′ − 1 = 0 8 7. (a) Draw a phase line diagram for the differential equation dx/dt = (2 − 5x)3(1 − 2x)(1 − 4x2). Expected in the diagram are equilibrium points and signs of x′ (or flow direction markers < and="">). (b) Draw a phase diagram using the phase line diagram of (a). Add these labels as appropriate: funnel, spout, node, stable, unstable. Show at least 10 threaded curves. A direction field is not required.