Engr 213

1 Introduction to Differential Equations Exercises 1.1 1. Second- ordain; one-dimensional. 2. Third-order; nonlinear because of (dy/dx)4 . 3. The di?erential comparability is ?rst-order. Writing it in the get to x(dy/dx) + y 2 = 1, we forebode that it is nonlinear in y because of y 2 . However, theme it in the form (y 2 ? 1)(dx/dy) + x = 0, we jut that it is linear in x. 4. The di?erential equation is ?rst-order. Writing it in the form u(dv/du) + (1 + u)v = ueu we see that it is linear in v. However, writing it in the form (v + uv ? ueu )(du/dv) + u = 0, we see that it is nonlinear in u. 5. Fourth-order; linear 6. Second-order; nonlinear because of romaine lettuce(r + u) 7. Second-order; nonlinear because of 1 + (dy/dx)2 8. Second-order; nonlinear because of 1/R2 9. Third-order; linear 10. Second-order; nonlinear because of x2 ? 11. From y = e?x/2 we let y = ? 1 e?x/2 . Then 2y + y = ?e?x/2 + e?x/2 = 0. 2 12. From y = 6 5 ? 6 e?20t we capture dy/dt = 24e?20t , so that 5 dy + 20y = 24e?20t + 20 dt 6 6 ?20t ? e 5 5 = 24. = 5e3x cos 2x ? 12e3x sin 2x, so that 13. From y = e3x cos 2x we reserve y = 3e3x cos 2x ? 2e3x sin 2x and y y ? 6y + 13y = 0. y = burn x + cos x ln(sec x + erythema solare x). Then y + y = tan x. 14. From y = ? cos x ln(sec x + tan x) we obtain y = ?1 + sin x ln(sec x + tan x) and 15.

Writing ln(2X ? 1) ? ln(X ? 1) = t and di?erentiating implicitly we obtain 2 dX 1 dX ? =1 2X ? 1 dt X ? 1 dt 2 1 ? 2X ? 1 X ? 1 dX =1 dt -4 -2 -2 -4 X 4 2 2 4 t 2X ? 2 ? 2X + 1 dX =1 (2X ? 1)(X ? 1) dt dX = ?(2X ? 1)(X ? 1) = (X ? 1)(1 ? 2X). dt Exponentiating both sides of the implicit resolvent we obtain 2X ? 1 et ? 1 = et =? 2X ? 1 = Xet ? et =? (et ? 1) = (et ? 2)X =! ? X = t . X ?1 e ?2 Solving et ? 2 = 0 we get t = ln 2. Thus, the result is de?ned on (??, ln 2) or on (ln 2, ?). The graph of the solution de?ned on (??, ln 2) is dashed, and the graph of the solution de?ned on (ln 2, ?) is solid. 1 Exercises 1.1 16. Implicitly di?erentiating the solution we obtain dy dy...If you want to get a full essay, order it on our website: OrderEssay.net

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