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MAT1801 Mathematics for Engineers I 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. 1 MAT1801 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) 2 MAT1801 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. 3 MAT1801