Elementary Calculus: Principle of Superposition

We conclude this section with a proof of Theorem 1. The proof uses the Principle of Superposition.

Suppose x(t) and y(t) are solutions of the two first order linear differential equations

Notice that all three differential equations have the same p(t). The Principle of Superposition follows from the Constant and Sum Rules for derivatives:

PROOF OF THEOREM 1

We are given that y and x are solutions of

(1)

y' + p(t)y = f(t)

and

(2)

x' + p(t)x = 0.

We must prove that a function u(t) is a solution of

(3)

u' + p(t)u = f(t)

if and only if u = y + Cx for some constant C.

Assume first that u = y + Cx. By the Principle of Superposition,

u' + p(t)u = f(t) + C · 0= f(t),

so u is a solution of (3).

Now assume that u is a solution of (3). Using the Principle of Superposition again,

(u - y)' + p(t)(u - y) = f(t) - f(t) = 0.

Thus u - y is a solution of the homogeneous linear equation (2). The general solution of equation (2) is Cx. Therefore for some constant C,

u - y = Cx

and

u = y + Cx.

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Principle of Superposition We conclude this section with a proof of Theorem 1. The proof uses the Principle of Superposition. Suppose x(t) and y(t) are solutions of the two first order linear differential equations x' + p(t)x = f(t), y' + p(t)y = g(t). Then for any constants A and B, the function u(t) = Ax(t) + By(t) is a solution of the linear differential equation u' + p(i)u = Af(t) + Bg(t). Notice that all three differential equations have the same p(t). The Principle of Superposition follows from the Constant and Sum Rules for derivatives: PROOF OF THEOREM 1 We are given that y and x are solutions of (1) y' + p(t)y = f(t) and (2) x' + p(t)x = 0. We must prove that a function u(t) is a solution of (3) u' + p(t)u = f(t) if and only if u = y + Cx for some constant C. Assume first that u = y + Cx. By the Principle of Superposition, u' + p(t)u = f(t) + C · 0= f(t), so u is a solution of (3). Now assume that u is a solution of (3). Using the Principle of Superposition again, (u - y)' + p(t)(u - y) = f(t) - f(t) = 0. Thus u - y is a solution of the homogeneous linear equation (2). The general solution of equation (2) is Cx. Therefore for some constant C, u - y = Cx and u = y + Cx.
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