Question

(1 point) For the differential equations solution to this equation through the point - 25 does the existence/uniqueness theorem guarantee that there is a ? . (-7,28)? 2 2. (-3,-5)? ?3. (-4,34)? | 4. (-5,5)? ?

Answer 1

consider the differential equation dy/dx = squareroot y ^2 - 25 By existence and uniqueness theorem f mod x, y = squareroot y ^2 - 25 df/dy = 1/2 squareroot y ^2 - 25 2 y = y/squareroot y ^2 - 25 The function f (x, y) is continuous if y ^2 - 25 greaterthanorequalto 0 y ^2 greaterthanorequalto 25 y greaterthanorequalto 5 or y lessthanorequalto - 5 y epsilon (- infinity, - 5] union [5, infinity) df/dy = y/squareroot y ^2 - 5 is continuous when y ^2 - 25 > 0 thus df/dy is continuous in the interval (- infinity, - 5) union (5, infinity) \r\n for the point (- 7, 28) x _ 0 = -7, y _ 0 = 28 thus y _ 0 = 28 epsilon (- infinity, - 5) union (5, infinity) therefore exestence and uniqueness theorem that the solution to this equation through the