Question

(a) Use the fact that 3x2 - y2 = c is a one-parameter family of solutions of the differential equation y dy = 3x to find an implicit solution of the initial-value problem y dy = 3x, y(2) = -7. 2 – 3x2 + 69 Then sketch the graph of the explicit solution of this problem. 20 20 20+ - ,2) (-7, 2) -20 * 10 -10 -10 10 20 -20 20 * 20 - 10 0 1 (2, -7) (2, -7) -20 -20 -20- -20F @x Give the interval I of definition of the explicit solution. (Enter your answer using interval notation.) (b) Are there any explicit solutions of y = 3x that pass through the origin? (If yes, enter the explicit solutions as a comma-separated list. If not, enter NONE.) y = $ 32 + 69

Answer 1

3x2-y?=c is one parameter family of solutions of differential eqn y dy = 3x Initial condition is y (2) = -7 Since 32²-4 2-c is the solution (one - parameter family) of yy' = 3x to find value of c, substitute x=2 y = -7 in 3x²-y²=ca = 3x 3(4) - (-7)2 = 0 C= 12-49 = -37 one solution is 3x² - y² = - 37 y = 322+37 y = 3x²+37 , y = -13x2+37 But y = 33x?+37 does not satisfy the given condition y (2) - -7. So this solution is Rejected, So Correct Sol" is y = -√3x²+37 Explicit sola is y = - +3x2+37 Explicit sol" is defined for 3x²+377,0 which is true for all xER.

Interval of definition of explicit sol" is (-00,00) y = -13x2+37 is a down ward parabola passing through (29-7) Ophon (3) of graph tracing is correct. ŠTo -10 -8 - -4 - 2T Ý posto a 12,-7) passes through No, there is No explicit sol" that ORIGIN.