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351 AMATH Homework 2
Due Oct 16(F)
Section2.1 14,16,19,31,40 Section2.4 3,12,15,29 Section2.6 8,21,23,24
Section 2.1
In each of Problems 13 through 20 nd the solution of the given initial value problem. 14. y + 2y = te2t , y (1) = 0 16. y + (2/t)y = (cos t)/t2 , y ( ) = 0, t > 0 19. t3 y + 4t2 y = et , y (1) = 0, t < 0 31. Consider the initial value problem
3 y y = 3t + 2et , 2 y (0) = y0 .
Find the value of y0 that separates solutions that grow positively as t from those that grow negatively. How does the solution that corresponds to this critical value of y0 behave as t ? In Problem 40 use the method of Problem 38 to solve the given dierential equation. 40. y + (1/t)y = 3 cos 2t, t>0
Section 2.4
In each of Problems 1 through 6 determine (without solving the problem) and interval in which the solution of the given initial value problem is certain to exist. 3. y + (tan t)y = sin t, y ( ) = 0 In each of Problems 7 through 12 state where in the ty -plane the hypotheses of Teorem 2.4.2 are satised. 12. dy = (cot t)y dt 1+y 1
In each of Problems 13 through 16 solve the given initial value problem and determine how the interval in which the solution exists depends on the initial value y0 . 15. y + y 3 = 0, y (0) y0 = Bernoulli Equations. Sometimes it is possible to solve a nonlinear equation by making a change of the dependent variable that converts it into a linear equation. The most important such equation has the form
y + p(t)y = q (t)y n .
and is called a Bernoulli equation after Jakob Bernoulli. Problem 27 through 31 deal with equations of this type. 29. y = ry ky 2 , r > 0 and k > 0. This equation is important in population dynamics and is discussed in detail in Section 2.5.
Section 2.6
Determin whether each of the equations in Problems 1 through 12 is exact. If it is exact, nd the solution. 8. (ex sin y + 3y )dx (3x ex sin y )dy = 0 Show that the equations in Problems 19 through 22 are not exact but become exact when multiplied by the given integrating factor. Then solve the equations. 21. ydx + (2x yey )dy = 0, (x, y ) = y 23. Show that if (Nx My )/M = Q, where Q is a function of y only, then the dierential equation
M + Ny = 0
has an integrating factor of the form
(y ) = exp
Q(y )dy.
24. Show that if (Nx My )/(xM yN ) = R, where R depends on the quantity xy only, then the dierential equation
M + Ny = 0
has an integrating factor of the form (xy ). Find a general formula for this integrating factor.
2

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Washington - MATH - 124

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