# Elementary Differential Equations and Boundary Value Problems - Boyce W.E.

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Section 4.3, Page 224

1. First solve the homogeneous D.E. The characteristic

68

Section 4.3

3 2

equation is r- r - r + 1 = 0, and the roots are r = -1,

1, 1; hence yc(t) = c—e-t + c2et + c3tet. Using the

superposition principle, we can write a particular solution as the sum of particular solutions corresponding to the D.E. y'^-y^-y'+y = 2e-t and y'^-y^-y'+y = 3. Our initial choice for a particular solution, Y—, of the

first equation is Ae-t; but e-t is a solution of the homogeneous equation so we multiply by t. Thus,

Y—(t) = Ate-t. For the second equation we choose

Y2(t) = B, and there is no need to modify this choice.

The constants are determined by substituting into the individual equations. We obtain A = 1/2, B = 3. Thus, the general solution is

y = c—e-t + c2et + c3tet + 3 + (te-t)/2.

4 2 2 2

5. The characteristic equation is r - 4r = r (r -4) = 0, so yc(t) = c— + c2t + c3e-2t + c4e2t. For the particular

2

solution correspnding to t we assume 22

Y— = t (At + Bt + C) and for the particular solution

corresponding to et we assume Y2 = Det. Substituting Y—, in the D.E. yields -48A = 1, B = 0 and 24A-8C = 0 and substituting Y2 yields -3D = 1. Solving for A, B, C and

D gives the desired solution.

9. The characteristic equation for the related homogeneous

D.E. is r3 + 4r = 0 with roots r = 0, +2i, -2i. Hence yc(t) = c— + c2cos2t + c3sin2t. The initial choice for

Y(t) is At + B, but since B is a solution of the homogeneous equation we must multiply by t and assume Y(t) = t(At+B). A and B are found by substituting in the D.E., which gives A = 1/8, B = 0, and thus the general solution is y(t) = c— + c2cos2t + c3sin2t + (1/8)t . Applying the I.C. we have y(0) = 0 ^ c— + c2 = 0, y'(0) = 0 ^ 2c3 = 0, and y"(0) = 1 ^ -4c2 + 1/4 = 1, which have the solution c1 = 3/16, c2 = -3/16, c3 = 0.

For small t the graph will approximate 3(1-cos2t)/16 and

for large t it will be approximated by t /8.

13. The characteristic equation for the homogeneous D.E. is

32

r - 2r + r = 0 with roots r = 0,1,1. Hence the complementary solution is yc(t) = c— + c2et + c3tet. We

Section 4.3

69

consider the differential equations y"' - 2y" + y' = t3

and y"' - 2y" + y' = 2et separately. Our initial choice for a particular solution, Y1, of the first equation is

32

A0t + A1t + A2t + A3; but since a constant is a solution of the homogeneous equation we must multiply by t. Thus

32

Y1(t) = t(A0t + A1t + A2t + A3). For the second equation

we first choose Y2(t) = Bet, but since both et and tet

are solutions of the homogeneous equation, we multiply by

2 2 t t to obtain Y2(t) = Bt e . Then Y(t) = Y1(t) + Y2(t) by

the superposition principle and y(t) = yc(t) + Y(t).

17. The complementary solution is yc(t) = c1 + c2e-t + c3et + c4tet. The superposition principle allows us to consider separately the D.E. yiv - y'" - y" +y' = t2 + 4 and yiv - y'" - y" + y' = tsint. For the first equation our

2

initial choice is Y1(t) = A0t + A1t + A2; but this must be multiplied by t since a constant is a solution of the

2

homogeneous D.E. Hence Y1(t) = t(A0t + A1t + A2). For

the second equation our initial choice that

Y2 = (B0t + B1)cost + + (C0t + C1)sint does not need to be modified. Hence 2

Y(t) = t(A0t + A1t + A2) + (B0t + B1)cost + (C0t + C1)sint.

2

20. (D-a)(D-b)f = (D-a)(Df-bf) = D f - (a+b)Df + abf and

2

(D-b)(D-a)f = (D-b)(Df-af) = D f - (b+a)Df + baf. Since

a+b = b+a and ab = ba, we find the given equation holds

for any function f.

22a. The D.E. of Problem 13 can be written as

D(D-1)2y = t3 + 2et. Since D4 annihilates t3 and (D-1)

annihilates 2et, we have D5(D-1) 3y = 0, which corresponds to Eq.(ii) of Problem 21. The solution of this equation

432

is y(x) = A1t + A2t + A3t + A4t + A5 +

2 -t -t (B1t + B2t + B3)e . Since A5 + (B2t + B3)e are

solutions of the homogeneous equation related to the original D.E., they may be deleted and thus

4 3 2 2 -t

Y(t) = A1t4 + A2t3 + A3t2 + A4t + B1t2e .

70

Section 4.4

22

22b. (D+1) (D +1) annihilates the right side of the D.E. of Problem 14.

3 2 2

22e. D (D +1) annihilates the right side of the D.E. of Problem 17.

Section 4.4, Page 229

1. The complementary solution is yc = c1 + c2cost + c3sint and thus we assume a particular solution of the form

Y = u1(t) + u2(t)cost + u3(t)sint. Differentiating and

assuming Eq.(5), we obtain Y' = -u2sint + u3cost and

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