1. (3pts) Find the general solution for the equation 2xy + y (No y' 4y4 + y2 need to write your solution in the explicit form.) 2. (3pts) Find the general solution of 2,4 = Express the general solution in the explicit form. 3. (4pts) Find the solution of the given initial value problem in explicit form: 3x2 2y – 3 1 y' =
Find the general solution of the differential equation xy' = y + (x2 + y2 y(4) = 3
Find the P.S. of the IVP: x2 + 2xy + y2 1+ (x + y)2 y(x = 0) = 4 Primes denote derivatives WRT X. (y'a
Show that if x and y are real numbers, x2 + y2 >= 2xy and (x + y)2 >= 4xy; When does equality hold (with proof)? Show that if x and y are real numbers, x2 + y2 2xy and (x y) 2Hry. When does equality hold (with proof)?
QUESTION 2 Calculate and simplify det if f(x,y) = O x2 - y2 + 2xy (x2 + y2)2 O 4x(+- x² + 3y2) (x² + y²)3 o 4x(x² – 3y?) (x² + y²)3 OO O x2 - y2 + 2xy (x² + y²)?
f(x, y) = x2 + y2 + 2xy + 6. 1- Find all the local extremas and 2) does the function f have an absolute max or min on R2
--- T2-1 (20 Points): Find the P.S. of the IVP: x2 + 2xy + y2 1+ (x + y)2 y(x = 0) = 4 Primes denote derivatives WRT X.
4. Find the general and particular solution to the following: y' = 2xy + 3x? eh, y(0) = 5.
T2-1 (20 Points): Find the P.S. of the IVP: x2 + 2xy + y2 1 + (x + y)2 y(x = 0) = 4 Primes denote derivatives WRT x. Sy' T2-2 (20 Points): Find the G.S. of the DE: xy' + y = 3x2 Prime denotes derivative WRT x. (Hint: guess a P.S. yı = Axa)
5. Let F = (3x2 + 2xy + y2)i + (x2 + 2xy + 4y3)j, and let C be any curve in R2 which begins at the origin and ends at (1, 2). Find ( F.dr.