This document summarizes key points about linear differential equations: 1. Students should check that theorems in this section about general solutions of linear equations reduce to those in the previous section for n=2. 2. Examples show finding linear combinations of solutions "by inspection" or trial and error to satisfy given equations. 3. Imposing initial conditions on general solutions yields particular solutions for various differential equations.

1. SECTION 5.2
GENERAL SOLUTIONS OF LINEAR EQUATIONS
Students should check each of Theorems 1 through 4 in this section to see that, in the case
n = 2, it reduces to the corresponding theorem in Section 5.1. Similarly, the computational
problems for this section largely parallel those for the previous section. By the end of Section
5.2 students should understand that, although we do not prove the existence-uniqueness theorem
now, it provides the basis for everything we do with linear differential equations.
The linear combinations listed in Problems 1-6 were discovered "by inspection" — that is, by
trial and error.
1. (5/2)(2x) + (-8/3)(3x2) + (-1)(5x - 8x2) = 0
2. (-4)(5) + (5)(2 - 3x2) + (1)(10 + 15x2) = 0
3. (1)(0) + (0)(sin x) + (0)(ex) = 0
4. (1)(17) + (-17/2)(2 sin2x) + (-17/3)(3 cos2x) = 0, because sin2x + cos2x = 1.
5. (1)(17) + (-34)(cos2x) + (17)(cos 2x) = 0, because 2 cos2x = 1 + cos 2x.
6. (-1)(ex) + (1)(cosh x) + (1)(sinh x) = 0, because cosh x = (ex + e-x)/2 and
sinh x = (ex - e-x)/2.
1 x x2
7. W = 0 1 2 x = 2 is nonzero everywhere.
0 0 2
ex e2 x e3 x
8. W = ex 2e 2 x 3e 3 x = 2 e 6 x is never zero.
ex 4e 2 x 9e 3 x
9. W = ex(cos2x + sin2x) = ex ≠ 0
10. W = x-7ex(x + 1)(x + 4) is nonzero for x > 0.
11. W = x3e2x is nonzero if x ≠ 0.
12. W = x-2[2 cos2(ln x) + 2 sin2(ln x)] = 2x-2 is nonzero for x > 0.
In each of Problems 13-20 we first form the general solution

2. y(x) = c1y1(x) + c2y2(x) + c3y3(x),
then calculate y′(x) and y″(x), and finally impose the given initial conditions to determine the
values of the coefficients c1, c2, c3.
13. Imposition of the initial conditions y (0) = 1, y ′(0) = 2, y ′′(0) = 0 on the general solution
y ( x ) = c1e x + c2e − x + c3e −2 x yields the three equations
c1 + c2 + c3 = 1, c1 − c2 − 2c3 = 2, c1 + c2 + 4c3 = 0
with solution c1 = 4 / 3, c2 = 0, c3 = − 1/ 3. Hence the desired particular solution is
given by y(x) = (4ex - e-2x)/3.
14. Imposition of the initial conditions y (0) = 0, y ′(0) = 0, y′′(0) = 3 on the general solution
y ( x ) = c1e x + c2e 2 x + c3e3 x yields the three equations
c1 + c2 + c3 = 1, c1 + 2c2 + 3c3 = 2, c1 + 4c2 + 9c3 = 0
with solution c1 = 3 / 2, c2 = − 3, c3 = 3 / 2. Hence the desired particular solution is
given by y(x) = (3ex - 6e2x + 3e3x)/2.
15. Imposition of the initial conditions y (0) = 2, y′(0) = 0, y′′(0) = 0 on the general solution
y ( x ) = c1e x + c2 x e x + c3 x 2 e3 x yields the three equations
c1 = 2, c1 + c2 = 0, c1 + 2c2 + 2c3 = 0
with solution c1 = 2, c2 = − 2, c3 = 1. Hence the desired particular solution is given by
y(x) = (2 - 2x + x2)ex.
16. Imposition of the initial conditions y (0) = 1, y ′(0) = 4, y ′′(0) = 0 on the general solution
y ( x ) = c1e x + c2e 2 x + c3 x e 2 x yields the three equations
c1 + c2 = 1, c1 + 2c2 + c3 = 4, c1 + 4c2 + 4c3 = 0
with solution c1 = − 12, c2 = 13, c3 = − 10. Hence the desired particular solution is
given by y(x) = -12ex + 13e2x - 10xe2x.
17. Imposition of the initial conditions y (0) = 3, y′(0) = − 1, y′′(0) = 2 on the general
solution y ( x ) = c1 + c2 cos 3x + c3 sin 3x yields the three equations

3. c1 + c2 = 3, 3c3 = − 1, − 9c2 = 2
with solution c1 = 29 / 9, c2 = − 2 / 9, c3 = − 1/ 3. Hence the desired particular solution is
given by y(x) = (29 - 2 cos 3x - 3 sin 3x)/9.
18. Imposition of the initial conditions y (0) = 1, y ′(0) = 0, y′′(0) = 0 on the general solution
y ( x) = e x (c1 + c2 cos x + c3 sin x ) yields the three equations
c1 + c2 = 1, c1 + c2 + c3 = 0, c1 + 2c3 = 0
with solution c1 = 2, c2 = − 1, c3 = − 1. Hence the desired particular solution is given by
y(x) = ex(2 - cos x - sin x).
19. Imposition of the initial conditions y (1) = 6, y ′(1) = 14, y ′′(1) = 22 on the general
solution y ( x) = c1 x + c2 x 2 + c3 x 3 yields the three equations
c1 + c2 + c3 = 6, c1 + 2c2 + 3c3 = 14, 2c2 + 6c3 = 22
with solution c1 = 1, c2 = 2, c3 = 3. Hence the desired particular solution is given by
y(x) = x + 2x2 + 3x3.
20. Imposition of the initial conditions y (1) = 1, y ′(1) = 5, y ′′(1) = − 11 on the general
solution y ( x ) = c1 x + c2 x −2 + c3 x −2 ln x yields the three equations
c1 + c2 = 1, c1 − 2c2 + c3 = 5, 6c2 − 5c3 = − 11
with solution c1 = 2, c2 = − 1, c3 = 1. Hence the desired particular solution is given by
y(x) = 2x - x-2 + x-2 ln x.
In each of Problems 21-24 we first form the general solution
y(x) = yc(x) + yp(x) = c1y1(x) + c2y2(x) + yp(x),
then calculate y′(x), and finally impose the given initial conditions to determine the values of
the coefficients c1 and c2.
21. Imposition of the initial conditions y (0) = 2, y′(0) = − 2 on the general solution
y ( x ) = c1 cos x + c2 sin x + 3 x yields the two equations c1 = 2, c2 + 3 = − 2 with
solution c1 = 2, c2 = − 5. Hence the desired particular solution is given by
y(x) = 2 cos x - 5 sin x + 3x.

4. 22. Imposition of the initial conditions y (0) = 0, y′(0) = 10 on the general solution
y ( x) = c1e 2 x + c2e −2 x − 3 yields the two equations c1 + c2 − 3 = 0, 2c1 − 2c2 = 10 with
solution c1 = 4, c2 = − 1. Hence the desired particular solution is given by
y(x) = 4e2x - e-2x - 3.
23. Imposition of the initial conditions y (0) = 3, y′(0) = 11 on the general solution
y ( x ) = c1e − x + c2 e3 x − 2 yields the two equations c1 + c2 − 2 = 3, − c1 + 3c2 = 11 with
solution c1 = 1, c2 = 4. Hence the desired particular solution is given by
y(x) = e-x + 4e3x - 2.
24. Imposition of the initial conditions y (0) = 4, y′(0) = 8 on the general solution
y ( x ) = c1e x cos x + c2 e x cos x + x + 1 yields the two equations c1 + 1 = 4, c1 + c2 + 1 = 8
with solution c1 = 3, c2 = 4. Hence the desired particular solution is given by
y(x) = ex(3 cos x + 4 sin x) + x + 1.
25. L[y] = L[y1 + y2] = L[y1] + L[y2] = f + g
26. (a) y1 = 2 and y2 = 3x (b) y = y1 + y2 = 2 + 3x
27. The equations
c1 + c2x + c3x2 = 0, c2 + 2c3x + 0, 2c3 = 0
(the latter two obtained by successive differentiation of the first one) evidently imply —
by substituting x = 0 — that c1 = c2 = c3 = 0.
28. If you differentiate the equation c0 + c1 x + c2 x 2 + + cn x n = 0 repeatedly, n times in
succession, the result is the system
c0 + c1 x + c2 x 2 + + cn x n = 0
c1 + 2c2 x + + ncn x n −1 = 0
(n − 1)!cn −1 + n !cn x = 0
n !cn = 0
of n+1 equations in the n+1 coefficients c0 , c1 , c2 , , cn . Upon substitution of
x = 0, the (k+1)st of these equations reduces to k !ck = 0, so it follows that all these
coefficients must vanish.

5. 29. If c0erx + c1xerx + ⋅⋅⋅ + cnxnerx = 0, then division by erx yields
c0 + c1x + ⋅⋅⋅ + cnxn = 0,
so the result of Problem 28 applies.
30. When the equation x2y″ - 2xy′ + 2y = 0 is rewritten in standard form
y″ + (-2/x)y′ + (2/x2)y = 0,
the coefficient functions p1(x) = -2/x and p2(x) = 2/x2 are not continuous at x = 0.
Thus the hypotheses of Theorem 3 are not satisfied.
31. (a) Substitution of x = a in the differential equation gives y′′(a) = − p y′(a) − q(a).
(b) If y(0) = 1 and y′(0) = 0, then the equation y″ - 2y′ - 5y = 0 implies that
y″(0) = 2y′(0) + 5y(0) = 5.
32. Let the functions y1, y2, ⋅⋅⋅ , yn be chosen as indicated. Then evaluation at
x = a of the (k - 1)st derivative of the equation c1y1 + c2y2 + ⋅⋅⋅ cnyn = 0 yields
ck = 0. Thus c1 = c2 = ⋅⋅⋅ = cn = 0, so the functions are linearly independent.
33. This follows from the fact that
1 1 1
a b c = (b − a)(c − b)(c − a).
a2 b2 c2
34. W(f1, f2, ⋅⋅⋅, fn) = V exp(rix), and neither V nor exp(rix) vanishes.