a) Given that dy = yx*dx State the order, type of the derivative, linearity and most appropriate method for finding its general solution (5pt) b) Solve for y (5pt) c) Modify equation of part a) so that it becomes a Linear equation (4pt) d) Match the graphs to all appropriate IVP choices (6pt) (a) y(1) = 1, y(1) = -2 (b) y(-1) =0, y'(-1) = -4 (e) y(1) = 1, y'(1) =2 (d) y(0) = -1, y'(0) = 2 (e) y(0) = -1, y'(0) =0 (f) y(0) = -4, y'(0) = -2 -5- - 5+a) Given that x2 af =f - xf dx Convert this problem into the standard form and solve for f b) Solve the following IVP (hint: think substitution) y' = (y -9x)- -7 y(0) =1 4Given that ([5xez + C) y= a) Identify integrating factor that was used to solve the original problem (4pt) b) Use information from part a) to reconstruct original Linear equation (8pt) y' + P(x)y = Q(x) c) What should be the value of & in order for the following equation to be Exact (2pt) (5x + ky)dx + (10x - 8y')dy = 0d) Use your value of k from part c) to verify the "exactness" and find general solution U (6pt)It is projected that someday human kind will be able to move to Mars. If that will happen successfully, then in the future First Credit Union of Mars will be created. Assuming that the FCUoM will offer a savings account with 10% annual interest and a $17 monthly maintenance fee. Please help us find the following: a) Equation that models Martian's savings account balance (8pt) b) Assuming that an account is opened with $10,000, how much money will it contain after 7 years? (6pt) c) If citizens of the Moon enjoy the same interest rate but have no maintenance fees on their savings accounts. How much money will a Lunarian' have in his/her account after 7 years? (assuming same initial deposit of $10,000). (6pt)Suppose that in 1620, a ship with 100 young Ukrainian pilgrims arrived at the shores of New England to establish a new settlement. Amongst them, there was an entrepreneurial carpenter who also happen to be well versed in Differential Equations. While crafting his business plan, clever carpenter assumed that the population of his newly established town will grow at the annual rate of approximately 3.8%. a) Model this growth with a differential equation (start from dP/dt = KP and derive the solution) (8pt) b) Assuming that the people of that town only used wooden chairs for sitting (and one person needs only one chair to sit), please help the carpenter verify his calculations and determine how many chairs will the town need in the year 1640? What about 1700? What about 2021? (6pt) c) Please also help the Mayor to determine at which rate population of that town will be changing in the year 1645? This information will really help the city council to determine, how many baby cribs town will need that year! (6pt)
