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

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In each of Problems 1 through 8 determine whether the given pair of functions is linearly

independent or linearly dependent.

1. f (t) = t2 + 5t, g(t) = t2 - 5t

2. f (9) = cos39, g(9) = 4cos3 9 - 3cos9

3. f (t) = ext cos /.it, g(t) = ext sinit, /ë = 0

4. f (x) = e3x, g(x) = e3(x-1)

5. f (t) = 3t - 5, g(t) = 9t - 15 6. f (t) = t, g(t) = t 1

7. f(t) = 3t, g(t) = |t| 8. f(x) = x3, g(x) =|x|3

9. The Wronskian of two functions is W (t) = t sin2 t. Are the functions linearly independent or linearly dependent? Why?

10. The Wronskian of two functions is W (t) = t2 - 4. Are the functions linearly independent or linearly dependent? Why?

11. If the functions y2 and y2 are linearly independent solutions of y" + p (t)y' + q (t)y = 0, prove that ñ2y2 and ñ2ó2 are also linearly independent solutions, provided that neither ñ2 nor ñ2 is zero.

12. If the functions y2 and y2 are linearly independent solutions of y" + p(t)y' + q (t)y = 0, prove that y3 = y2 + y2 and y4 = y2 - y2 also form a linearly independent set of solutions. Conversely, if y3 and y4 are linearly independent solutions of the differential equation, show that y1 and y2 are also.

13. If the functions y2 and y2 are linearly independent solutions of y" + p(t)y' + q (t)y = 0, determine under what conditions the functions y3 = a2y2 + a2y2 and y4 = b2y2 + b2y2 also form a linearly independent set of solutions.

14. (a) Prove that any two-dimensional vector can be written as a linear combination of i + j and i-j.

(b) Prove that if the vectors x = x2i + x2j and y = y2i + y2j are linearly independent, then any vector z = Zji + z2j can be expressed as a linear combination of x and y. Note that if x and y are linearly independent, then x2y2 - x2y2 = 0. Why?

In each of Problems 15 through 18 find the Wronskian of two solutions of the given differential

equation without solving the equation.

15. 12y" - t (t + 2)y' + (t + 2)y = 0 16. (cos t)y" + (sint)y' - ty = 0

17. x2y" + xy' + (x2 - v2)y = 0, Bessel’s equation

18. (1 - x2)yw - 2xy + a(a + 1)y = 0, Legendre’s equation

19. Show that if p is differentiable and p(t) > 0, then the Wronskian W (t) of two solutions of [p(t)y']' + q (t)y = 0 is W(t) = ñ/p(t), where ñ is a constant.

20. If y2 and y2 are linearly independent solutions of ty" + 2y' + te‘y = 0 and if

W (Ó ³, Ó2)(1) = 2, find the value of W (y2, y2)(5).

21. If y2 and y2 are linearly independent solutions of t2y" - 2y' + (3 + t)y = 0 and if

W (y2, y2)(2) = 3, find the value of W (y2, y2)(4).

22. If the Wronskian of any two solutions of y" + p(t)y' + q (t)y = 0 is constant, what does this imply about the coefficients p and q ?

23. If f, g, and h are differentiable functions, show that W (fg, fh) = f2 W (g, h).

In Problems 24 through 26 assume that p and q are continuous, and that the functions y1 and y2

are solutions of the differential equation y" + p(t)y' + q (t)y = 0 on an open interval I.

3.4 Complex Roots ofthe Characteristic Equation

153

24. Prove that if y1 and y2 are zero at the same point in I, then they cannot be a fundamental set of solutions on that interval.

25. Prove that if y1 and y2 have maxima or minima at the same point in I, then they cannot be a fundamental set of solutions on that interval.

26. Prove that if y2 and y2 have a common point of inflection t0 in I, then they cannot be a fundamental set of solutions on I unless both p and q are zero at t0.

27. Show that 1 and 12 are linearly independent on -I < 1 < I; indeed, they are linearly independent on every interval. Show also that W (t, t2) is zero at t = 0. What can you conclude from this about the possibility that t and t2 are solutions of a differential equation y" + p(t)y + q (1)y = 0? Verify that 1 and 12 are solutions of the equation 12y" - 2ty + 2y = 0. Does this contradict your conclusion? Does the behavior of the Wronskian of t and t2 contradict Theorem 3.3.2?

28. Show that the functions f (t) = 12|t | and g(t) = t3 are linearly dependent on 0 < t < ³ and on -1 < t < 0, but are linearly independent on -1 < t < I. Although f and g are linearly independent there, show that W (f, g) is zero for all t in -1 < t < I. Hence f and g cannot be solutions of an equation y" + p(t)y' + q (t)y = 0 with p and q continuous on

-I < t < I.

3.4 Complex Roots of the Characteristic Equation

We continue our discussion of the equation

ay" + by' + ñó = 0, (1)

where a, b, and ñ are given real numbers. In Section 3.1 we found that if we seek solutions of the form y = ert, then r must be a root of the characteristic equation

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