Section 25–1 Linear differential equations

(Linear operator / Independent solutions / Forced solution)

 

In this section, Feynman discusses linear operator, independent solutions (free solutions) of a homogeneous differential equation (right-hand side of the equation is zero), and the forced solution of an inhomogeneous differential equation.

 

1. Linear operator:

“We sometimes call this an operator notation, but it makes no difference what we call it, it is just ‘shorthand’ (Feynman et al., 1963, section 25–1 Linear differential equations).”

 

Feynman calls L an operator notation and says that it makes no difference what we call it. He provides two important statements: (1) L(x+y) = L(x) + L(y), and (2) for constant a, L(ax) = aL(x). However, we can call L a linear operator instead of operator notation. To be specific, linear operators are defined with two necessary conditions: (1) For x and y Î V, L(x+y) = L(x) + L(y) (L is additive), and (2) For x Î V, a Î R, L(ax) = aL(x) (L is homogeneous) in which V is a real vector space and R is a set of real numbers. Simply put, a linear operator provides an operation or instruction that informs us how we should do with x and y that may be numbers, functions, or vectors. Perhaps Feynman should problematize the word linear and explain that it is not simply about straight lines.

 

Feynman mentions that there may be more derivatives and more terms in L in more complicated problems. If the two conditions for a linear operator are maintained, then such a problem is a linear problem. In solving any linear problems, we can combine two inputs such as the velocity of an object in a train and the velocity of the train, that will result in the sum of their respective outputs. On the other hand, a differential equation such as ẍ² + x = 0 is a non-linear problem because it has a square term that violates the two conditions. In general, many problems in fluid dynamics, atmospheric physics, and general relativity are based on nonlinear equations that are unsolvable or difficult to be solved.

 

2. Independent solutions:

“It turns out that the number of what we call independent solutions that we have obtained for our oscillator problem is only two (Feynman et al., 1963, section 25–1 Linear differential equations).”

 

Feynman explains that there are only two independent solutions if we have a second-order differential equation. He adds that the number of independent solutions in the general case depends upon what is called the number of degrees of freedom. However, we could obtain the general solution of a second-order differential equation, e.g., mẍ + kx = 0, simply by using two integrations. That is, the general solution can be expressed as x = Ax₁(t) + Bx₂(t) in which A and B are dependent on the initial conditions. More importantly, the general solution in terms of two independent solutions x₁(t) and x₂(t) can be related to the principle of superposition, but this is discussed in the next section.

 

In a footnote, Feynman states that “solutions which cannot be expressed as linear combinations of each other are called independent.” Specifically, one may prefer the phrase “linearly independent solutions” and explain it using two vectors and two functions. In general, two vectors or two functions are linearly independent if one of them cannot be expressed as a multiple of the other. For example, two vectors x and 2x are linearly dependent because we can have 2x = 2 ´ x or 2(x). On the contrary, x and x² are linearly independent because x² is not a constant multiple of x. Similarly “moving in the x–direction” and “moving in the y–direction” are linearly independent in the sense that we cannot replace the x–direction by y–direction, or vice versa.

 

3. Forced solution:

“Therefore, to the “forced” solution we can add any “free” solution, and we still have a solution. (Feynman et al., 1963, section 25–1 Linear differential equations).”

 

Feynman explains that the “forced” solution does not die out because it is driven by a force. Ultimately, the general solution is almost equal to the “force” solution as the “free” solution slowly becomes negligible. Formally speaking, the free “solution” is the complementary function and the “forced” solution is the particular integral of the second-order differential equation. One should also explain the three constants that appear in the general solution. In the “free” solution, any amplitude (or arbitrary constant) is possible, but the two arbitrary constants are dependent on how the system was started. On the other hand, the constant or amplitude of the “forced solution” is not arbitrary because it depends on the “forcing” function.

 

Feynman shows that L(x_J + x₁) = F(t) + 0 = F(t) and says that we can add any “free” solution to the “forced” solution and is still a solution. It is worthwhile to distinguish three different principles of superposition. First, L(x+y) = 0 + 0 = 0: “Let L be any linear operator. Then if y = u and y = v are both solutions of L(y) = 0, the same is true of y = c₁u + c₂v, for any constants c₁ and c₂ (Sokolnikoff & Redheffer, 1966, p. 171).” Second, L(x+y) = F(t) + 0 = F(t): “Let u be a particular solution of L(y) = f, where L is any linear operator, and let v satisfy the homogeneous equation L(y) = 0. Then y = u + v satisfies L(y) = f, and every solution of L(y) = f can be obtained in this way (Ibid, p. 183).” Third, L(x+y) = F₁(t) + F₂(t): “Let y₁ satisfy the equation L(y₁) = f₁ and let y₂ satisfy L(y₂) = f₂, where L is any linear operator. Then, for any constants c₁ and c₂, the function y = c₁y₁ + c₂y₂ satisfies L(y) = c₁f₁ + c₂f₂ (Ibid, p. 186).” For consistency’s sake, Sokolnikoff and Redheffer’s use of the symbol T is changed to L.

 

Questions for discussion:

1. How would you define a linear operator?

2. How would you explain the independent solutions of a second-order differential equation are linearly independent? 

3. How would you explain the forced solution will become a steady solution?

 

The moral of the lesson: we can combine two independent solutions to form a “free” solution, and we can combine the “free” solution with a “forced” solution to form a general solution (using two slightly different principles of superposition).

 

References:

1. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

2. Sokolnikoff, I. S., & Redheffer, R. M. (1966). Mathematics of Physics and Modern Engineering (2nd Ed.). Singapore: McGraw-Hill.