Higher-Order Linear Differential Equations
Linear Homogeneous Differential Equations with Constant Coefficients
CS 130: Mathematical Methods in Computer Science Ordinary Differential Equations Nestine Hope S. Hernandez Algorithms and Complexity Laboratory Department of Computer Science University of the Philippines, Diliman nshernandez@dcs.upd.edu.ph Day 12
Higher-Order Linear Differential Equations
Linear Homogeneous Differential Equations with Constant Coefficients
Ordinary Differential Equations
Higher-Order Linear Differential Equations Solutions of a Linear Differential Equation
Linear Homogeneous Differential Equations with Constant Coefficients
Higher-Order Linear Differential Equations
Linear Homogeneous Differential Equations with Constant Coefficients
Ordinary Differential Equations
Higher-Order Linear Differential Equations Solutions of a Linear Differential Equation
Linear Homogeneous Differential Equations with Constant Coefficients
Higher-Order Linear Differential Equations
Linear Homogeneous Differential Equations with Constant Coefficients
The General Linear Equation
The general linear differential equation of order n is an equation that can be written
Higher-Order Linear Differential Equations
Linear Homogeneous Differential Equations with Constant Coefficients
The General Linear Equation
The general linear differential equation of order n is an equation that can be written b0 (x)
dn y dn−1 y dy + bn (x)y = R(x). + b (x) + · · · + bn−1 (x) 1 dxn dxn−1 dx
Higher-Order Linear Differential Equations
Linear Homogeneous Differential Equations with Constant Coefficients
The General Linear Equation
The general linear differential equation of order n is an equation that can be written dn y dn−1 y dy + bn (x)y = R(x). + b (x) + · · · + bn−1 (x) 1 dxn dxn−1 dx If the value of the function R(x) is zero for all x, then the equation is called a homogeneous linear differential equation. Otherwise, we call it a nonhomogeneous linear differential equation. b0 (x)
Higher-Order Linear Differential Equations
Linear Homogeneous Differential Equations with Constant Coefficients
Differential Operators In calculus, differentiation is often denoted by D, that is,
dy = Dy. dx
Higher-Order Linear Differential Equations
Linear Homogeneous Differential Equations with Constant Coefficients
Differential Operators dy In calculus, differentiation is often denoted by D, that is, = Dy. dx The symbol D is called a differential operator because it transforms a differentiable function into another function.
Higher-Order Linear Differential Equations
Linear Homogeneous Differential Equations with Constant Coefficients
Differential Operators dy In calculus, differentiation is often denoted by D, that is, = Dy. dx The symbol D is called a differential operator because it transforms a differentiable function into another function. Polynomial expressions involving D, such as D + 3, D3 + 2D2 − D + 5, and 2x3 D4 − x2 D2 + D − 1, are also differential operators.
Higher-Order Linear Differential Equations
Linear Homogeneous Differential Equations with Constant Coefficients
Differential Operators dy In calculus, differentiation is often denoted by D, that is, = Dy. dx The symbol D is called a differential operator because it transforms a differentiable function into another function. Polynomial expressions involving D, such as D + 3, D3 + 2D2 − D + 5, and 2x3 D4 − x2 D2 + D − 1, are also differential operators. In general, we define an nth -order differential operator or polynomial operator to be L = an (x)Dn + an−1 (x)Dn−1 + · · · + a1 (x)D + a0 (x).
Higher-Order Linear Differential Equations
Linear Homogeneous Differential Equations with Constant Coefficients
Differential Equations Any linear differential equation can be expressed in terms of the operator D.
Higher-Order Linear Differential Equations
Linear Homogeneous Differential Equations with Constant Coefficients
Differential Equations Any linear differential equation can be expressed in terms of the operator D. Example: y 00 + 5y 0 + 6y = 5x − 3
Higher-Order Linear Differential Equations
Linear Homogeneous Differential Equations with Constant Coefficients
Differential Equations Any linear differential equation can be expressed in terms of the operator D. Example: y 00 + 5y 0 + 6y = 5x − 3 =⇒ D2 y + 5Dy + 6y = 5x − 3
Higher-Order Linear Differential Equations
Linear Homogeneous Differential Equations with Constant Coefficients
Differential Equations Any linear differential equation can be expressed in terms of the operator D. Example: y 00 + 5y 0 + 6y = 5x − 3 =⇒ D2 y + 5Dy + 6y = 5x − 3 =⇒ (D2 + 5D + 6)y = 5x − 3
Higher-Order Linear Differential Equations
Linear Homogeneous Differential Equations with Constant Coefficients
Superposition Principle for Homogeneous Equations If y1 , y2 , · · · , yk are solutions of the homogeneous linear differential equation of order n b0 (x)y (n) + b1 (x)y (n−1) + · · · + bn−1 (x)y 0 + bn (x)y = 0 then y = c1 y1 + c2 y2 + · · · ck yk where the ci s are arbitrary constants, is also a solution to the equation.
Higher-Order Linear Differential Equations
Linear Homogeneous Differential Equations with Constant Coefficients
Superposition Principle for Homogeneous Equations If y1 , y2 , · · · , yk are solutions of the homogeneous linear differential equation of order n b0 (x)y (n) + b1 (x)y (n−1) + · · · + bn−1 (x)y 0 + bn (x)y = 0 then y = c1 y1 + c2 y2 + · · · ck yk where the ci s are arbitrary constants, is also a solution to the equation. That is, any linear combination of solutions of a homogeneous linear differential equation is also a solution.
Higher-Order Linear Differential Equations
Linear Homogeneous Differential Equations with Constant Coefficients
Superposition Principle for Homogeneous Equations If y1 , y2 , · · · , yk are solutions of the homogeneous linear differential equation of order n b0 (x)y (n) + b1 (x)y (n−1) + · · · + bn−1 (x)y 0 + bn (x)y = 0 then y = c1 y1 + c2 y2 + · · · ck yk where the ci s are arbitrary constants, is also a solution to the equation. That is, any linear combination of solutions of a homogeneous linear differential equation is also a solution.
Example Find the unique solution of the initial value problem x2 y 00 + 2xy 0 − 12y = 0,
y(1) = 4,
y 0 (1) = 5.
Higher-Order Linear Differential Equations
Linear Homogeneous Differential Equations with Constant Coefficients
Superposition Principle for Homogeneous Equations If y1 , y2 , · · · , yk are solutions of the homogeneous linear differential equation of order n b0 (x)y (n) + b1 (x)y (n−1) + · · · + bn−1 (x)y 0 + bn (x)y = 0 then y = c1 y1 + c2 y2 + · · · ck yk where the ci s are arbitrary constants, is also a solution to the equation. That is, any linear combination of solutions of a homogeneous linear differential equation is also a solution.
Example Find the unique solution of the initial value problem x2 y 00 + 2xy 0 − 12y = 0,
y(1) = 4,
note: x3 and x−4 are solutions to the ODE
y 0 (1) = 5.
Higher-Order Linear Differential Equations
Linear Homogeneous Differential Equations with Constant Coefficients
Linearly Independent Solutions of a Linear Differential Equation
Higher-Order Linear Differential Equations
Linear Homogeneous Differential Equations with Constant Coefficients
Linearly Independent Solutions of a Linear Differential Equation
Definition Suppose each of the functions f1 (x), f2 (x), · · · least n − 1 derivatives. The determinant
f1 f2
0
f10 f 2
. . W (f1 , f2 , · · · , fn ) =
. .
. .
(n−1) (n−1)
f f 1
2
is called the Wronskian of the functions.
, fn (x) possesses at
··· ···
···
fn fn0 . . . (n−1)
fn
,
Higher-Order Linear Differential Equations
Linear Homogeneous Differential Equations with Constant Coefficients
Linearly Independent Solutions of a Linear Differential Equation
Definition Suppose each of the functions f1 (x), f2 (x), · · · least n − 1 derivatives. The determinant
f1 f2
0
f10 f 2
. . W (f1 , f2 , · · · , fn ) =
. .
. .
(n−1) (n−1)
f f 1
2
, fn (x) possesses at
··· ···
···
fn fn0 . . . (n−1)
fn
,
is called the Wronskian of the functions.
Theorem Let {y1 , y2 , · · · , yn } be n solutions of the homogeneous linear equation on an interval I. Then the set of solutions is linearly independent on I if and only if W (y1 , y2 , · · · , yn ) 6= 0 for every x in the interval.
Higher-Order Linear Differential Equations
Linear Homogeneous Differential Equations with Constant Coefficients
General Solution of a Homogeneous Equation Let {y1 , y2 , · · · , yn } be a linearly independent set of solutions of the homogeneous linear equation b0 (x)y (n) + b1 (x)y (n−1) + · · · + bn−1 (x)y 0 + bn (x)y = 0 on an interval I. Then the general solution of the equation on the interval is y = c1 y1 + c2 y2 + · · · + cn yn where the ci s are arbitrary constants.
Higher-Order Linear Differential Equations
Linear Homogeneous Differential Equations with Constant Coefficients
General Solution of a Nonhomogeneous Equation Let yp be any particular solution of the nonhomogeneous linear nth-order differential equation on an interval I, and let {y1 , y2 , · · · , yn } be a linearly independent set of solutions of the associated homogeneous differential equation on I. Then the general solution of the equation on the interval is y = c1 y1 + c2 y2 + · · · + cn yn + yp where the ci s are arbitrary constants.
Higher-Order Linear Differential Equations
Linear Homogeneous Differential Equations with Constant Coefficients
General Solution of a Nonhomogeneous Equation Let yp be any particular solution of the nonhomogeneous linear nth-order differential equation on an interval I, and let {y1 , y2 , · · · , yn } be a linearly independent set of solutions of the associated homogeneous differential equation on I. Then the general solution of the equation on the interval is y = c1 y1 + c2 y2 + · · · + cn yn + yp where the ci s are arbitrary constants.
That is, the general solution of a nonhomogeneous equation is then y = yc + yp where yc is called the complementary function (the general solution to the associated homogeneous differential equation) and yp is any particular solution of the nonhomogeneous differential equation.
Higher-Order Linear Differential Equations
Linear Homogeneous Differential Equations with Constant Coefficients
Example 1. Find the general solution of y 00 = 4. 2. Find the general solution of y 00 − y = 4.
Higher-Order Linear Differential Equations
Linear Homogeneous Differential Equations with Constant Coefficients
Ordinary Differential Equations
Higher-Order Linear Differential Equations Solutions of a Linear Differential Equation
Linear Homogeneous Differential Equations with Constant Coefficients
Higher-Order Linear Differential Equations
Linear Homogeneous Differential Equations with Constant Coefficients
Homogeneous Equations: auxiliary eqn w/ distinct real roots
Example 1.
d3 y dx3
2
d y − 4 dx 2 +
dy dx
+ 6y = 0
Higher-Order Linear Differential Equations
Linear Homogeneous Differential Equations with Constant Coefficients
Homogeneous Equations: auxiliary eqn w/ distinct real roots
Example 1.
d3 y dx3
2
d y dy − 4 dx 2 + dx + 6y = 0 auxiliary equation: m3 − 4m2 + m + 6 = 0
Higher-Order Linear Differential Equations
Linear Homogeneous Differential Equations with Constant Coefficients
Homogeneous Equations: auxiliary eqn w/ distinct real roots
Example 1.
d3 y dx3
2
d y dy − 4 dx 2 + dx + 6y = 0 auxiliary equation: m3 − 4m2 + m + 6 = 0 roots: m = −1, 2, 3
Higher-Order Linear Differential Equations
Linear Homogeneous Differential Equations with Constant Coefficients
Homogeneous Equations: auxiliary eqn w/ distinct real roots
Example 1.
d3 y dx3
2
d y dy − 4 dx 2 + dx + 6y = 0 auxiliary equation: m3 − 4m2 + m + 6 = 0 roots: m = −1, 2, 3 general solution:
y = c1 e−x + c2 e2x + c3 e3x
Higher-Order Linear Differential Equations
Linear Homogeneous Differential Equations with Constant Coefficients
Homogeneous Equations: auxiliary eqn w/ distinct real roots
Example 1.
d3 y dx3
2
d y dy − 4 dx 2 + dx + 6y = 0 auxiliary equation: m3 − 4m2 + m + 6 = 0 roots: m = −1, 2, 3 general solution:
y = c1 e−x + c2 e2x + c3 e3x 3
2
d y d y dy 2. 3 dx 3 + 5 dx2 − 2 dx = 0
Higher-Order Linear Differential Equations
Linear Homogeneous Differential Equations with Constant Coefficients
Homogeneous Equations: auxiliary eqn w/ distinct real roots
Example 1.
d3 y dx3
2
d y dy − 4 dx 2 + dx + 6y = 0 auxiliary equation: m3 − 4m2 + m + 6 = 0 roots: m = −1, 2, 3 general solution:
y = c1 e−x + c2 e2x + c3 e3x 3
2
d y d y dy 2. 3 dx 3 + 5 dx2 − 2 dx = 0
3.
d2 y dx2
− 4y = 0
Higher-Order Linear Differential Equations
Linear Homogeneous Differential Equations with Constant Coefficients
Homogeneous Equations: auxiliary eqn w/ distinct real roots
Example 1.
d3 y dx3
2
d y dy − 4 dx 2 + dx + 6y = 0 auxiliary equation: m3 − 4m2 + m + 6 = 0 roots: m = −1, 2, 3 general solution:
y = c1 e−x + c2 e2x + c3 e3x 3
2
d y d y dy 2. 3 dx 3 + 5 dx2 − 2 dx = 0
3.
d2 y dx2
− 4y = 0
Higher-Order Linear Differential Equations
Linear Homogeneous Differential Equations with Constant Coefficients
Homogeneous Equations: auxiliary eqn w/ repeated real roots
Example 1.
d4 y dx4
3
2
d y d y dy − 7 dx 3 + 18 dx2 − 20 dx + 8y = 0
Higher-Order Linear Differential Equations
Linear Homogeneous Differential Equations with Constant Coefficients
Homogeneous Equations: auxiliary eqn w/ repeated real roots
Example 1.
d4 y dx4
3
2
d y d y dy − 7 dx 3 + 18 dx2 − 20 dx + 8y = 0 auxiliary equation: m4 − 7m3 + 18m2 − 20m + 8 = 0
Higher-Order Linear Differential Equations
Linear Homogeneous Differential Equations with Constant Coefficients
Homogeneous Equations: auxiliary eqn w/ repeated real roots
Example 1.
d4 y dx4
3
2
d y d y dy − 7 dx 3 + 18 dx2 − 20 dx + 8y = 0 auxiliary equation: m4 − 7m3 + 18m2 − 20m + 8 = 0 roots: m = 1, 2, 2, 2
Higher-Order Linear Differential Equations
Linear Homogeneous Differential Equations with Constant Coefficients
Homogeneous Equations: auxiliary eqn w/ repeated real roots
Example 1.
d4 y dx4
3
2
d y d y dy − 7 dx 3 + 18 dx2 − 20 dx + 8y = 0 auxiliary equation: m4 − 7m3 + 18m2 − 20m + 8 = 0 roots: m = 1, 2, 2, 2 general solution:
y = c1 ex + c2 e2x + c3 xe2x + c4 x2 e2x
Higher-Order Linear Differential Equations
Linear Homogeneous Differential Equations with Constant Coefficients
Homogeneous Equations: auxiliary eqn w/ repeated real roots
Example 1.
d4 y dx4
3
2
d y d y dy − 7 dx 3 + 18 dx2 − 20 dx + 8y = 0 auxiliary equation: m4 − 7m3 + 18m2 − 20m + 8 = 0 roots: m = 1, 2, 2, 2 general solution:
y = c1 ex + c2 e2x + c3 xe2x + c4 x2 e2x
2.
d4 y dx4
3
d y + 2 dx 3 +
d2 y dx2
=0
Higher-Order Linear Differential Equations
Linear Homogeneous Differential Equations with Constant Coefficients
Homogeneous Equations: auxiliary eqn w/ imaginary roots
Example 1.
d3 y dx3
2
d y dy − 3 dx 2 + 9 dx + 13y = 0
Higher-Order Linear Differential Equations
Linear Homogeneous Differential Equations with Constant Coefficients
Homogeneous Equations: auxiliary eqn w/ imaginary roots
Example 1.
d3 y dx3
2
d y dy − 3 dx 2 + 9 dx + 13y = 0 auxiliary equation: m3 − 3m2 + 9m + 13 = 0
Higher-Order Linear Differential Equations
Linear Homogeneous Differential Equations with Constant Coefficients
Homogeneous Equations: auxiliary eqn w/ imaginary roots
Example 1.
d3 y dx3
2
d y dy − 3 dx 2 + 9 dx + 13y = 0 auxiliary equation: m3 − 3m2 + 9m + 13 = 0 roots: m = −1, 2 ± 3i
Higher-Order Linear Differential Equations
Linear Homogeneous Differential Equations with Constant Coefficients
Homogeneous Equations: auxiliary eqn w/ imaginary roots
Example 1.
d3 y dx3
2
d y dy − 3 dx 2 + 9 dx + 13y = 0 auxiliary equation: m3 − 3m2 + 9m + 13 = 0 roots: m = −1, 2 ± 3i general solution:
y = c1 e−x + c2 e2x cos 3x + c3 e2x sin 3x
Higher-Order Linear Differential Equations
Linear Homogeneous Differential Equations with Constant Coefficients
Homogeneous Equations: auxiliary eqn w/ imaginary roots
Example 1.
d3 y dx3
2
d y dy − 3 dx 2 + 9 dx + 13y = 0 auxiliary equation: m3 − 3m2 + 9m + 13 = 0 roots: m = −1, 2 ± 3i general solution:
y = c1 e−x + c2 e2x cos 3x + c3 e2x sin 3x
2.
d4 y dx4
2
d y + 8 dx 2 + 16y = 0
Higher-Order Linear Differential Equations
Linear Homogeneous Differential Equations with Constant Coefficients
Exercises 1. y 00 + y 0 − 2y = 0 2. y 00 − 10y 0 + 50y = 0 3. y 00 − 8y 0 + 15y = 0 4. 4y (4) − 45y 00 − 70y 0 − 24y = 0 5. 4y (4) − 4y 000 − 23y 00 + 12y 0 + 36y = 0 6. y (5) + y (4) − 7y 000 − 11y 00 − 8y 0 − 12y = 0
Higher-Order Linear Differential Equations
Linear Homogeneous Differential Equations with Constant Coefficients
Questions? See you next meeting!