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10. f1(t) = 2t - 3, f2(t) = t3 + 1, f3(t) = 2t2 - t, f4(t) = t2 + t + 1
In each of Problems 11 through 16 verify that the given functions are solutions of the differential equation, and determine their Wronskian.
11. /" + Ó = 0; 1, cos t, sin t
12. yv + Ó = 0; 1, t, cos t, sin t
13. Ó'+ 2/- /- 2y = 0; et, e-t, e-2t
14. yv + 2Ó + Ó = 0; 1, t, e-t, te-t
15. xy - Ó = 0; 1, x, x3
16. x3Ó + x2Ó - 2xy + 2y = 0; x, x2, 1 /x
17. Show that W(5, sin2 t, cos2t) = 0 for all t. Can you establish this result without direct evaluation of the Wronskian?
18. Verify that the differential operator defined by
L [y] = y(n) + P1 (t)y(n-1) + ¦¦¦ + pn (t)y
is a linear differential operator. That is, show that
L [c1 y + c2 y2] = c1 L [y1] + c2 L
4.1 General Theory of nth Order Linear Equations
where y1 and y2 are n times differentiable functions and c1 and c2 are arbitrary constants. Hence, show that if y1, y2,•••, yn are solutions of L[y] = 0, then the linear combination c1 y + ¦¦¦ + c„y„ is also a solution of L [y] = 0.
19. Let the linear differential operator L be defined by
where a0, a1, • • • , a„ are real constants.
(a) Find L[tn]•
(b) Find L[ert],
(c) Determine four solutions of the equation yv — 5y" + 4y = 0. Do you think the four solutions form a fundamental set of solutions? Why?
20. In this problem we show how to generalize Theorem 3.3.2 (Abel’s theorem) to higher order equations. We first outline the procedure for the third order equation
Hint: The derivative of a 3-by-3 determinant is the sum of three 3-by-3 determinants obtained by differentiating the first, second, and third rows, respectively.
(b) Substitute for yj", yH, and Ó" from the differential equation; multiply the first row by p3, the second row by p2, and add these to the last row to obtain
for this case.
In each of Problems 21 through 24 use Abel’s formula (Problem 20) to find the Wronskian of a fundamental set of solutions of the given differential equation.
25. The purpose of this problem is to show that if W(y1, • • •, y„)(t0) = 0 for some t0 in an interval I, then y1, • • • , yn are linearly independent on I, and if they are linearly independent and solutions of
L [y] = a0 y(n) + a1 y(n—1) + ¦¦¦ + a„y,
Ó + p1 (t) Ó' + p2 (t)y + p3 (t)y = 0 •
Let y1, y2, and y3 be solutions of this equation on an interval I.
(a) If W = W(y1, y2, y3), show that
y ó2 y3
W' = y1 Ó2 Ó3
y y y
W' = —p^OW^
(c) Show that
It follows that W is either always zero or nowhere zero on I.
(d) Generalize this argument to the nth order equation
y(„) + p1(t) y(n—1) + ¦¦¦ + pn (t) y = 0
with solutions ó , • • , y„. That is, establish Abel’s formula,
21. Ó" + 2ó — Ó — 3y = 0
23. ty" + 2y" — Ó+ ty = 0
22. yv + y = 0
24. t2 ylv + ty" + Ó — 4y = 0
L [ y] = y(n) + p1 (t) y(n—1) + ¦¦¦ + p„ (t) y = 0 on I, then W(y1, • • •, yn) is nowhere zero in I.
Chapter 4. Higher Order Linear Equations
(a) Suppose that W(y1,..., yn)(t0) = 0, and suppose that
C1 yl(t) + -" + ñïÓï (´) = 0 (ii)
for all t in I. By writing the equations corresponding to the first n - 1 derivatives ofEq. (ii) at t0, show that Cj = ••• = cn = 0. Therefore, y1,yn are linearly independent.
(b) Suppose that yl,..., yn are linearly independent solutions of Eq. (i). If W (yl,..., yn )(t0) = 0forsome t0, show that there is a nonzero solution ofEq. (i) satisfying the initial conditions
y(t0) = y(t0) = •••= y(n-X)(t0) = 0.
Since y = 0 is a solution of this initial value problem, the uniqueness part of Theorem 4.1.1 yields a contradiction. Thus W is never zero.
26. Show that if ó is a solution of
Ó" + pi (t)f + p2(t) Ó + p3(t) y = 0,
then the substitution y = y (t)v(t) leads to the following second order equation for V:
ylv"' + (3y + p1 yl)v" + (3Ó1 + 2 px y1 + p2 y1)v' = 0.