# MAT1801 Mathematics for Engineers I by byrnetown70

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Second Order Ordinary Diﬀerential Equations

1. Linear Second Order Equations

A linear second order diﬀerential equation is one that can be written in the
form
a2 (x)y + a1 (x)y + a0 (x)y = f (x).
If f (x) ≡ 0, then this equation is homogeneous. Otherwise, it is called a nonhomo-
geneous equation.

We shall be focusing on the special case where the coeﬃcients a0 (x), a1 (x) and
a2 (x) are constant:
ay + by + cy = f (x),                               (1)
with a = 0.

Theorem (Existence and Uniqueness) For any real numbers a, b, c, α and
β, there exists a unique function y(x) that is a solution of equation (1) and satisﬁes
the initial conditions y(x0 ) = α and y (x0 ) = β.

2. Linear Independence

Deﬁnition Let y1 and y2 be functions of x. For any constants c1 and c2 , the
function y = c1 y1 + c2 y2 is said to be a linear combination of y1 and y2 . The linear
combination is said to be trivial if c1 = 0 and c2 = 0. Otherwise, it is said to be
nontrivial.

Deﬁnition The functions y1 and y2 are said to be linearly dependent if there is
a nontrivial linear combination of y1 and y2 which is equal to zero (i.e. if each is a
constant multiple of the other). Otherwise, they are said to be linearly independent.

Deﬁnition The Wronskian of y1 and y2 is the function

W (y1 , y2 ) = y1 y2 − y1 y2 .

Theorem The functions y1 and y2 are linearly independent if and only if
W (y1 , y2 ) = 0 for all x.

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3. The Homogeneous Equation

Consider the homogeneous equation
ay + by + cy = 0,                                   (2)
with a = 0. If r is a root of the associated auxiliary equation
ax2 + bx + c = 0
then y = erx is a solution of equation (2). (Verify!)

Now, it is not hard to see that if y1 and y2 are both solutions of equation (2),
then any linear combination of y1 and y2 is also a solution of the equation.

Theorem If y1 and y2 are linearly independent solutions of equation (2), then
for any real numbers α and β, there exist unique constants c1 and c2 such that
c1 y1 + c2 y2 is a solution of (2) with the initial conditions y(x0 ) = α and y (x0 ) = β.

The function y = c1 y1 + c2 y2 is called the general solution of equation (2).

The following cases may occur, corresponding to the three types of quadratic
equation.

Case 1. Distinct real roots: if r and s are roots of the auxiliary equation, then
y1 = erx and y2 = esx are two linearly independent solutions of (2).

Case 2. Equal roots: if r is a double root of the auxiliary equation, then y1 = erx
and y2 = xerx are two linearly independent solutions of (2).

Case 3. Complex roots: if the auxiliary equation has complex roots α ± iβ,
then y1 = eαx cos βx and y2 = eαx sin βx are two linearly independent solutions of
(2).

4. The Nonhomogeneous Equation

Let us return to the nonhomogeneous equation (1). We have the following
Superposition Principle:

Theorem Let u1 be a solution of the equation
ay + by + cy = f1 (x),                                (3)
and let u2 be a solution of the equation
ay + by + cy = f2 (x).                                (4)

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Then, for any constants d1 and d2 , the function d1 u1 + d2 u2 is a solution of the
equation
ay + by + cy = d1 f1 (x) + d2 f2 (x).                     (5)

Now, suppose y1 and y2 are linearly independent solutions of the homogeneous
equation corresponding to equation (1) (i.e. the same equation with f (x) = 0).
Suppose further that yp is a particular solution of (1).

Then by the Superposition Principle, the function
y = c1 y1 + c2 y2 + yp                                (6)
is a solution of equation (1), for all real numbers c1 and c2 . In fact, the following
theorem tells us that this is the general solution.

Theorem Suppose yp is a particular solution of (1), and y1 and y2 are linearly
independent solutions of the associated homogeneous equation. Then, for any real
numbers α and β, there exist unique constants c1 and c2 such that the function in
(6) is a solution to the initial value problem
ay + by + cy = f (x);         y(x0 ) = α, y (x0 ) = β.

5. The Method of Undetermined Coeﬃcients

Consider the equation
ay + by + cy = f (x)
where f (x) is a sum of terms, each of which is a product of polynomials, exponential
functions and sine or cosine functions. To ﬁnd a particular solution:

(i) for every term in f of the type Pn (x)eαx , where Pn (x) is a polynomial of
degree n, use the form
yp = xs An xn + An−1 xn−1 + · · · + A1 x + A0 eαx
where   s = 0 if α is not a root of the associated auxiliary equation;
s = 1 if α is a simple root of the associated auxiliary equation;
s = 2 if α is a double root of the associated auxiliary equation.

(ii) for every term in f of the type Pn (x)eαx cos βx or Pn (x)eαx sin βx, where
Pn (x) is a polynomial of degree n, use the form
yp = xs (An xn + · · · + A0 ) eαx cos βx + xs (Bn xn + · · · + B0 ) eαx sin βx
where   s = 0 if α + iβ is not a root of the associated auxiliary equation;
s = 1 if α + iβ is a root of the associated auxiliary equation.

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