Cs130 day12

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!