SEC.. Separable Equations In each of problems 1 through 8 solve the given differential equation : ü 1. y ' x y x y, y 0 fl y - x 0 fl y - x 0 fl y - x3 3 c, y 0 ü. y ' x ^ y 1 + x 3 x y 1 + x 3, y 0 fl x 1 + x 3 - y 0 fl 1 3 ln 1 + x3 - y c, y 0, x -1 ü 3. y ' + y sin x 0 -y sin x fl sin x -y - fl-cos x y -1 + c, " y 0; also y 0 everywhere.
Untitled-3 ü 4. y ' 3 x - 1 3 + y 3 x - 1 3 + y, y -3 fl 3 + y 3 x - 1 fl 3 y + y x 3 - x + c, y -3 ü 7. y ' x - -x y + x - -x y + y, y + y 0 fl y + y x - -x fl y + y x + -x + c, y + y 0 ü 8. y ' x 1 + y x 1 + y fl x 1 + y fl x3 3 y + y3 3 + c In each of problems 9 through 0 : (a) Find the solution of the given initial value problem in explicit form. (b) Plot the graph of the solution. (c) Determine (at least approximately) the interval in which the solution is defined.
Untitled-3 3 ü 9. y ' 1 - x y, y 0-1 6 (a) 1 - x y fl y - 1 - x fl-y -1 x - x + c Since y 0-1 6 fl 6 c 1 fl y x - x - 6, x - x - 6 0 (b) 0.6 0.4 0. -4-4 6 8-0. -0.4-0.6 (c) x - x - 6 0 ñ x - 3 x + 0 ñ x 3orx - fl the solution is defined on -, - -, 3 3,.
4 Untitled-3 ü 10. y ' 1 - x y, y 1 - (a) 1 - x y fl y x - x + c Since y 1 -, fl c fl y - x - x + 1 fl y 1 - -1, take y - - x - x + 1 (b) - -1 1 3-0.5-1.0-1.5 -.0 (c) Since - x - x + 1 0 fl x - x + 1 0 fl-1 x fl the solution is defined on -1,.
Untitled-3 5 ü 11. x x + y -x y 0, y 0 1 (a) x + y -x 0 fl x x + y 0 fl x x - x + y c Since y 0 1, fl c - 1 fl x x - x + y -1 fl y -1-x x + x fl y -1 - x x + x fl y 0-1 + 1, take y -1 - x x + x (b) 1.0 0.8 0.6 0.4 0. -.0-1.5-1.0-0.5 0.5 1.0 (c) Since -1 - x x + x 0, and x -1.67835, x 0.768039 are the solutions of -1 - x x + x 0 by Newton's method fl the solution is defined on -1.67835, 0.768039.
6 Untitled-3 ü 1. r qr q, r 1 (a) r q r q, q 0 flq -1 q r - r fl ln q -r -1 + c Since r 1 fl c 1 fl r -1 1 - ln q fl r,1-ln q 0 1 - ln q (b) 5-4 - 4-5 -10 (c) Since 1 - ln q 0ñln q 1 ñq exp 1 º 1.6487 flthe solution is defined on -, -exp 1 -exp 1, 0 0, exp 1 exp 1,.
Untitled-3 7 ü 13. y ' x y + x y, y 0 - (a) x y + x y x y 1 + x fl x 1 + x y fl ln 1 + x y + c Since y 0 - fl c - fl ln 1 + x y - fl y ln 1 + x + 4, since y 0 -, take y - ln 1 + x + 4 (b) -.0 -. -.4 -.6 -.8-3.0-6 -4-4 6 (c) Since ln 1 + x + 4 0, " x œ flthe solution is defined on.
8 Untitled-3 ü 14. y ' xy 3 1 + x -1, y 0 1 (a) xy3 1 + x -1 fl y -3 x 1 + x -1 fl -1 y- 1 + x 1 + c Since y 0 1 fl c -3 fl y - - 1 + x 1 + 3 fl y 1-1 + x 1 + 3 ; since y 0 1, take y 1-1 + x 1 + 3, - 1 + x 1 + 3 > 0 (b) 4 3 1-1.5-1.0-0.5 0.0 0.5 1.0 1.5 (c) Since - 1 + x 1 + 3 > 0ñ3 > 1 + x 1 ñ 1 + x 1 < 3 ñx < 5 4ñ - 5 4 < x < 5 4 º 1.11803 flthe solution is defined on -1.11803, 1.11803.
Untitled-3 9 30. Consider the equation y x y - 4 x x - y. (i) ü (a) Show that it can be rewritten as y x y x - 4 1 - y x ; (ii) thus it is homogeneous. pf y - 4 x x - y x y x - 4 x 1 - y x y x - 4 1 - y x, x 0 ü (b) Introduce a new dependent variable v so that v y x, or y xv x. Express y x in terms of x, v, and v x. sol Let v y x ñ y xv x, then xv x v x + x v ü (c) Replace y and y x in y x y x - 4 1 - y x by the expressions from part (b) that involve v and v x. Show that the resulting differential equation is v + x v x v - 4 1 - v or x v x v - 4 1 - v. (iii) Observe that Eq.(iii) is separable. sol y x - 4 1 - y x v ñ v x + x v - 4 1 - v ñ x v v - 4 1 - v ü (d) Solve Eq. (iii), obtaining v implicitly in terms of x. sol x v v - 4 1 - v fl x -1 1 - v v - 4 v fl ln x - 1 4 ln - v - 3 ln + v + c 4 fl ln x + 1 4 ln - v + 3 ln + v c 4 fl ln x - v 1 4 + v 3 4 c 1
10 Untitled-3 ü (e) Find the solution of Eq. (i) by replacing v by y x in the solution in part (d). sol fl ln x - v 1 4 + v 3 4 ln x - y x 1 4 + y x 3 4 c 1 fl x - y x 1 4 + y x 3 4 c fl x 4 - y x + y x 3 x - y x + y 3 c ü (f) Draw a direction field and some integral curves for Eq. (i). Recall that the right side of Eq. (i) actually depends only on the ratio y x. This means that integral curves have the same slope at all points on any given straight line through the origin, although the slope changes from one line to another. Therefore the direction filed and the integral curves are symmetric with respect to the origin. Is this symmetry property evident from your plot? direction field
Untitled-3 11 integral curves for Eq. (i) Yes, the direction filed and the integral curves are symmetric with respect to the origin. ü 31. y x x + xy + y x (a) Show that the given equation is homogeneous. x + xy + y 1 + y x x + y x, x 0 (b) Solve the differential equation. Let v y x ñ y xv x, then v + x v 1 + v + v fl x v 1 + v fl x -1 1 1 + v v fl ln x arctan v + c arctan y x + c 1 fl x exp arctan y x + c 1
1 Untitled-3 (c) Draw a direction field and some integral curves. Are they symmetric with respect the origin? direction field
Untitled-3 13 integral curves Yes, the direction filed and the integral curves are symmetric with respect to the origin. ü 3. y x x + 3 y xy (a) Show that the given equation is homogeneous. x + 3 y xy x y + 3 y x 1 y x + 3 y x, x 0 (b) Solve the differential equation. Let v y x ñ y xv x, then v + x v 1 v + 3 v fl x v 1 + v v fl x -1 v 1 + v v fl ln x ln 1 + v + c ln 1 + y x + c 1 fl ln x 1 + y x - -1 c 1 fl x 1 + y x - -1 c
14 Untitled-3 (c) Draw a direction field and some integral curves. Are they symmetric with respect the origin? direction field
Untitled-3 15 integral curves Yes, the direction filed and the integral curves are symmetric with respect to the origin. ü 33. y x 4 y - 3 x x - y (a) Show that the given equation is homogeneous. 4 y x - 3 - y x, x 0
16 Untitled-3 (b) Solve the differential equation. Let v y x ñ y xv x, then v + x v fl x v fl x -1 4 v - 3 - v 4 v - 3 - v - v v + v - 3 v - v v + v - 3 - v fl ln x 1 4 ln -3-1 + v - 5 4 ln 3 3 + v + c 1 fl ln x 4 ln 3 v - 1 - ln 3 3 + v 5 + c fl ln x 4 ln 3 v - 1 - ln 3 3 + v 5 + c fl ln x 4 - ln 3 x -1 y - 1 + ln 3 3 + x -1 y 5 c fl ln x 4 3-1 x -1 y - 1-1 3 5 3 + x -1 y 5 c fl ln 3 4 y - x -1 3 x + y 5 c fl 3 4 y - x -1 3 x + y 5 c fl 3 4 3 x + y 5 c y - x
Untitled-3 17 (c) Draw a direction field and some integral curves. Are they symmetric with respect the origin? direction field
18 Untitled-3 integral curves Yes, the direction filed and the integral curves are symmetric with respect to the origin. ü 34. y x - 4 x + 3 y x + y (a) Show that the given equation is homogeneous. - 4 x + 3 y x + y - 4 + 3 y x, x 0 + y x
Untitled-3 19 (b) Solve the differential equation. Let v y x ñ y xv x, then v + x v fl x v fl x -1-4 + 3 v + v - 4 + 3 v + v - v + v -4-5 v - v v -4-5 v - v + v fl ln x - 3 ln -4 - v - 1 3 ln 1 + v + c 1 fl ln x 3 -ln -4 - v - ln 1 + v + c 1 fl ln x 3 + ln -4 - v + ln 1 + v c 1 fl ln x 3-4-v 1 + v c 1 fl ln x 3-4-x -1 y 1 + x -1 y c 1 fl ln -4 x - y x + y c 1 fl -4 x - y x + y c
0 Untitled-3 (c) Draw a direction field and some integral curves. Are they symmetric with respect the origin? direction field
Untitled-3 1 integral curves Yes, the direction filed and the integral curves are symmetric with respect to the origin.