Problems 9. Given that x, x2, and 1/x are solutions of the homogeneous equation corresponding to In cach of Problems 1 through 4, use the method of variation of parameters to determine the general solution of the given differential equation. xy"+x?y"-2xy +2y 2x, x> 0, 1. y +y tan t, determine a particular solution. 10. Find a formula involving integrals for a particular solution of the differential equation 2. y"-y' t 3. y-2y"-y'+2y e 4. y-y"+y'-y e sint y"-y"+y-y= g(1). In each of Problems 5 and 6, find the general solution of the given differential equation. Leave your answer in terms of one or more integrals. 11. Find a formula involving integrals for a particular solution of th differential equation es: Reducti Undeter y4) - y= g(1). 5. y-y+y-y-secr, -<1< 2 Hint: The functions sin t, cost, sinh t, and cosh set of solutions of the homogeneous equation. Mechan form a fundamen 6. y"-y= csc t, 0

For problems 1-3, use the Picard-Lindelöf Theorem to describe whether the given differential equation is guaranteed to have a solution and, if a solution is guaranteed to exist, whether any solution is guaranteed to be unique. Explain your reasoning. Problem 1. x'(t) = t² cos , x(0) = 0

This is a practice question from my Differential Equations course. How’d they get from the given equation to the answer? Textbook is very unclear; so I’m hoping for more detail and less skipping of steps… Thank you for your assistance in understanding this.

(c) Describe a scenario in which you are modeling the growth rate of a population, but the differential equations in the previous two parts (the Malthusian Growth Law and Lo- gistic Growth Law respectively) might not be appropriate. Then give a differential equation which models your sce- nario. Be creative, and don't use the equation we used in class!