1. 2.10 Solution by Variation
of Parameters
Section p1

2. We continue our discussion of nonhomogeneous linear ODEs, that is
2.10 Solution by Variation of Parameters
We continue our discussion of nonhomogeneous linear ODEs, that is
(1) y” + p(x)y’ + q(x)y = r(x).
To obtain yp when r(x) is not too complicated, we can often use the method of undetermined coefficients.
However, since this method is restricted to functions r(x) whose derivatives are of a form similar to r(x) itself (powers, exponential functions, etc.), it is desirable to have a method valid for more general ODEs (1), which we shall now develop.
It is called the method of variation of parameters and is credited to Lagrange (Sec. 2.1). Here p, q, r in (1) may be variable (given functions of x), but we assume that they are continuous on some open interval I.
Section p2

3. on I, and W is the Wronskian of y1, y2.
2.10 Solution by Variation of Parameters
Lagrange’s method gives a particular solution yp of (1) on I in the form
(2)
where y1, y2 form a basis of solutions of the corresponding homogeneous ODE
(3) y” + p(x)y’ + q(x)y = 0
on I, and W is the Wronskian of y1, y2.
(4) W = y1y2’ − y2y1’ (see Sec. 2.6).
Section p3

4. EXAMPLE 1 Method of Variation of Parameters
2.10 Solution by Variation of Parameters
EXAMPLE 1
Method of Variation of Parameters
Solve the nonhomogeneous ODE
Solution. A basis of solutions of the homogeneous ODE on any interval is y1 = cos x, y2 = sin x. This gives the Wronskian
W( y1, y2) = cos x cos x − sin x (−sin x) = 1.
From (2), choosing zero constants of integration, we get the particular solution of the given ODE
(Fig. 70)
Section p4

5. EXAMPLE 1 (continued) Solution. (continued 1)
2.10 Solution by Variation of Parameters
EXAMPLE 1 (continued)
Solution. (continued 1)
Figure 70 shows yp and its first term, which is small, so that x sin x essentially determines the shape of the curve
of yp. From yp and the general solution yh = c1y1 + c2y2 of the homogeneous ODE we obtain the answer
y = yh + yp = (c1 + ln|cos x|) cos x + (c2 + x) sin x.
Had we included integration constants −c1, c2 in (2),
then (2) would have given the additional
c1cos x + c2sin x = c1y1 + c2y2, that is, a general solution of the given ODE directly from (2). This will always be the case.
Section p5

6. EXAMPLE 1 (continued) Solution. (continued 2)
2.10 Solution by Variation of Parameters
EXAMPLE 1 (continued)
Solution. (continued 2)
Section p6

7. SUMMARY OF CHAPTER 2 Second-Order ODEs
Section 2.Summary p7

8. (1) y” + p(x)y’ + q(x)y = r(x) (Sec. 2.1).
SUMMARY OF CHAPTER 2 Second-Order ODEs
Second-order linear ODEs are particularly important in applications, for instance, in mechanics (Secs. 2.4, 2.8) and electrical engineering (Sec. 2.9). A second-order ODE is called linear if it can be written
(1) y” + p(x)y’ + q(x)y = r(x) (Sec. 2.1).
(If the first term is, say, f(x)y”, divide by f(x) to get the “standard form” (1) with y” as the first term.) Equation (1) is called homogeneous if r(x) is zero for all x considered, usually in some open interval; this is written r ≡ 0. Then
(2) y” + p(x)y’ + q(x)y = 0
Equation (1) is called nonhomogeneous if r(x) ≡ 0 (meaning r(x) is not zero for some x considered).
Section 2.Summary p8

9. (3) y(x0) = K0, y’(x0) = K1 (x0, K0, K1 given numbers; Sec. 2.1).
SUMMARY OF CHAPTER 2 Second-Order ODEs
(continued 1)
For the homogeneous ODE (2) we have the important superposition principle (Sec. 2.1) that a linear combination y = ky1 + ly2 of two solutions y1, y2 is again a solution.
Two linearly independent solutions y1, y2 of (2) on an open interval I form a basis (or fundamental system) of solutions on I, and y = c1y1 + c2y2 with arbitrary constants c1, c2 a general solution of (2) on I. From it we obtain a particular solution if we specify numeric values (numbers) for c1 and c2, usually by prescribing two initial conditions
(3) y(x0) = K0, y’(x0) = K (x0, K0, K1 given numbers; Sec. 2.1).
(2) and (3) together form an initial value problem. Similarly for (1) and (3).
Section 2.Summary p9

10. For a nonhomogeneous ODE (1) a general solution is of the form
SUMMARY OF CHAPTER 2 Second-Order ODEs
(continued 2)
For a nonhomogeneous ODE (1) a general solution is of the form
(4) y = yh + yp (Sec. 2.7).
Here yh is a general solution of (2) and yp is a particular solution of (1). Such a yp can be determined by a general method (variation of parameters, Sec. 2.10) or in many practical cases by the method of undetermined coefficients.
The latter applies when (1) has constant coefficients p and q, and r(x) is a power of x, sine, cosine, etc. (Sec. 2.7). Then we write (1) as
(5) y” + ay’ + by = r(x) (Sec. 2.7).
The corresponding homogeneous ODE y’ + ay’ + by = 0 has solutions y = eλx where λ is a root of
(6) λ2 + aλ + b = 0.
Section 2.Summary p10

11. Hence there are three cases (Sec. 2.2):
SUMMARY OF CHAPTER 2 Second-Order ODEs
(continued 3)
Hence there are three cases (Sec. 2.2):
Here ω* is used since ω is needed in driving forces.
Important applications of (5) in mechanical and electrical engineering in connection with vibrations and resonance are discussed in Secs. 2.4, 2.7, and 2.8.
Section 2.Summary p11

12. SUMMARY OF CHAPTER 2 Second-Order ODEs
(continued 4)
Another large class of ODEs solvable “algebraically” consists of the Euler–Cauchy equations
(7) x2y” + axy’ + by = 0 (Sec. 2.5).
These have solutions of the form y = xm, where m is a solution of the auxiliary equation
(8) m2 + (a − 1)m + b = 0.
Existence and uniqueness of solutions of (1) and (2) is discussed in Secs. 2.6 and 2.7, and reduction of order
in Sec. 2.1.
Section 2.Summary p12

13. 3.1 Homogeneous Linear ODEs
Section 3.1 p13

14. A second-order ODE is called linear if it can be written
3.1 Homogeneous Linear ODEs
A second-order ODE is called linear if it can be written
(1) y” + p(x)y’ + q(x)y = r(x)
and nonlinear if it cannot be written in this form.
Section 3.1 p14

15. (1) y(n) + pn−1(x)y(n−1) + … + p1(x)y’ + p0(x)y = r(x).
3.1 Homogeneous Linear ODEs
Recall from Section 1.1:
An ODE is of nth order if the nth derivative of y(n) = dny/dxn the unknown function y(x) is the highest occurring derivative. Thus the ODE is of the form
F(x, y, y’, … , y(n)) = 0
where lower order derivatives and y itself may or may not occur. Such an ODE is called linear if it can be written
(1) y(n) + pn−1(x)y(n−1) + … + p1(x)y’ + p0(x)y = r(x).
The coefficients p0, … , pn−1, and the function r on the right are any given functions of x, and y is unknown. y(n) has a coefficient 1. We call this the standard form.
An nth-order ODE that cannot be written in the form (1) is called nonlinear.
Section 3.1 p15

16. (2) y(n) + pn−1(x)y(n−1) + … + p1(x)y’ + p0(x)y = 0.
3.1 Homogeneous Linear ODEs
Recall from Section 1.1:
If r(x) is identically zero, r(x) ≡ 0 (zero for all x considered, usually on some open interval I), then (1) becomes
(2) y(n) + pn−1(x)y(n−1) + … + p1(x)y’ + p0(x)y = 0.
and is called homogeneous.
If r(x) is not identically zero, then the ODE is called
nonhomogeneous.
A solution of an nth-order (linear or nonlinear) ODE on some open interval I is a function y = h(x) that is defined and n times differentiable on I and is such that the ODE becomes an identity if we replace the unknown function y and its derivatives by h and its corresponding derivatives.
Section 3.1 p16

17. Homogeneous Linear ODEs: Superposition Principle. General Solution
The basic superposition or linearity principle of Sec. 2.1 extends to nth order homogeneous linear ODEs as follows.
Section 3.1 p17

18. Theorem 1 Fundamental Theorem for the Homogeneous Linear ODE (2)
3.1 Homogeneous Linear ODEs
Theorem 1
Fundamental Theorem
for the Homogeneous Linear ODE (2)
For a homogeneous linear ODE (2), sums and constant multiples of solutions on some open interval I are again solutions on I.
(This does not hold for a nonhomogeneous or nonlinear ODE!)
Section 3.1 p18

19. Definition General Solution, Basis, Particular Solution
3.1 Homogeneous Linear ODEs
Definition
General Solution, Basis, Particular Solution
A general solution of (2) on an open interval I is a solution of (2) on I of the form
(3) y(x) = c1y1(x) + … + cnyn(x) (c1, … , cn arbitrary) where y1, … , yn is a basis (or fundamental system) of solutions of (2) on I; that is, these solutions are linearly independent on I, as defined below.
A particular solution of (2) on I is obtained if we assign specific values to the n constants c1, … , cn in (3).
Section 3.1 p19

20. Definition Linear Independence and Dependence
3.1 Homogeneous Linear ODEs
Definition
Linear Independence and Dependence
Consider n functions y1(x), … , yn(x) defined on some interval I.
These functions are called linearly independent on I if the equation
(4) k1y1(x) + … + knyn(x) = 0 on I
implies that all k1, … , kn are zero. These functions are called linearly dependent on I if this equation also holds on I for some k1, … , kn not all zero.
Section 3.1 p20

21. 3.1 Homogeneous Linear ODEs
If and only if y1, … , yn are linearly dependent on I, we can express (at least) one of these functions on I as a “linear combination” of the other n − 1 functions, that is, as a sum of those functions, each multiplied by a constant (zero or not).
This motivates the term “linearly dependent.” For instance, if (4) holds with k1 ≠ 0, we can divide by k1 and express y1 as the linear combination
Note that when n = 2, these concepts reduce to those defined in Sec. 2.1.
Section 3.1 p21

22. Initial Value Problem. Existence and Uniqueness
3.1 Homogeneous Linear ODEs
Initial Value Problem. Existence and Uniqueness
An initial value problem for the ODE (2) consists of (2) and n initial conditions
(5) y(x0) = K0, y’(x0) = K1, …, y(n−1)(x0) = Kn−1
with given x0 in the open interval I considered, and given K0, …, Kn−1.
Section 3.1 p22

23. Theorem 2 Existence and Uniqueness Theorem for Initial Value Problems
3.1 Homogeneous Linear ODEs
Theorem 2
Existence and Uniqueness Theorem
for Initial Value Problems
If the coefficients p0(x), … , pn−1(x) of (2) are continuous on some open interval I and x0 is in I, then the initial value problem (2), (5) has a unique solution on I.
Section 3.1 p23

24. EXAMPLE 4 Initial Value Problem
3.1 Homogeneous Linear ODEs
EXAMPLE 4
Initial Value Problem
for a Third-Order Euler–Cauchy Equation
Solve the following initial value problem on any open interval I on the positive x-axis containing x = 1.
x3y”’ − 3x2y” + 6xy’ − 6y = 0,
y(1) = 2, y’(1) = 1, y”(1) 4.
Solution. (See next slide.)
Section 3.1 p24

25. m(m − 1)(m − 2)xm − 3m(m − 1)xm + 6mxm − 6xm = 0.
3.1 Homogeneous Linear ODEs
EXAMPLE 4 (continued)
Solution.
Step 1. General solution. As in Sec. 2.5 we try y = xm. By differentiation and substitution,
m(m − 1)(m − 2)xm − 3m(m − 1)xm + 6mxm − 6xm = 0.
Dropping xm and ordering gives m3 − 6m2 + 11m − 6 = 0. If we can guess the root m = 1, we can divide by m − 1 and find the other roots 2 and 3, thus obtaining the solutions x, x2, x3, which are linearly independent on I (see Example 2). [In general, one shall need a root-finding method, such as Newton’s (Sec. 19.2), also available in a CAS (Computer Algebra System).] Hence a general solution is
y = c1x + c2 x2 + c3 x3
valid on any interval I, even when it includes x = 0 where the coefficients of the ODE divided by x3 (to have the standard form) are not continuous.
Section 3.1 p25

26. Step 2. Particular solution.
3.1 Homogeneous Linear ODEs
EXAMPLE 4 (continued)
Solution. (continued)
Step 2. Particular solution.
The derivatives are y’ = c1 + 2c2x + 3c3x2 and y” = 2c2 + 6c3x. From this, and y and the initial conditions, we get by setting x = 1
(a) y(1) = c1 + c2 + c3 = 2
(b) y’(1) = c1 + 2c2 + 3c3 = 1
(c) y”(1) = c2 + 6c3 = −4
This is solved by Cramer’s rule (Sec. 7.6), or by elimination, which is simple, as follows: (b) − (a) gives
(d) c2 + 2c3 = −1. Then (c) − 2(d) gives c3 = −1.
Then (c) gives c2 = 1. Finally c1 = 2 from (a).
Answer: y = 2x + x2 − x3.
Section 3.1 p26

27. Linear Independence of Solutions. Wronskian
3.1 Homogeneous Linear ODEs
Linear Independence of Solutions. Wronskian
Linear independence of solutions is crucial for obtaining general solutions. Although it can often be seen by inspection, it would be good to have a criterion for it. Now Theorem 2 in Sec. 2.6 extends from order n = 2 to any n. This extended criterion uses the Wronskian W of n solutions y1, … , yn defined as the nth-order determinant
(6)
Note that W depends on x since y1, … , yn do. The criterion states that these solutions form a basis if and only if W is not zero; more precisely:
Section 3.1 p27

28. Theorem 3 Linear Dependence and Independence of Solutions
3.1 Homogeneous Linear ODEs
Theorem 3
Linear Dependence and Independence of Solutions
Let the ODE (2) have continuous coefficients p0(x), … , pn−1(x) on an open interval I. Then n solutions y1, … , yn of (2) on I are linearly dependent on I if and only if their Wronskian is zero for some x = x0 on I. Furthermore, if W is zero for x = x0, then W is identically zero on I. Hence if there is an x1 on I at which W is not zero, then y1, … , yn are linearly independent on I, so that they form a basis of solutions of (2) on I.
Section 3.1 p28

29. Theorem 4 A General Solution of (2) Includes All Solutions
3.1 Homogeneous Linear ODEs
A General Solution of (2) Includes All Solutions
Theorem 4
Existence of a General Solution
If the coefficients p0(x), … , pn−1(x) of (2) are continuous on some open interval I, then (2) has a general solution on I.
Section 3.1 p29

30. Theorem 5 General Solution Includes All Solutions
3.1 Homogeneous Linear ODEs
Theorem 5
General Solution Includes All Solutions
If the ODE (2) has continuous coefficients p0(x), … , pn−1(x) on some open interval I, then every solution y = Y(x) of (2) on I is of the form
(9) Y(x) = C1y1(x) + … + Cnyn(x)
where y1, … , yn is a basis of solutions of (2) on I and C1, … , Cn are suitable constants.
Section 3.1 p30