Linear Homogeneous Differential Equations with Constant Coefficients Linear Nonhomogeneous Differential Equations
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 13
Linear Homogeneous Differential Equations with Constant Coefficients Linear Nonhomogeneous Differential Equations
Ordinary Differential Equations
Linear Homogeneous Differential Equations with Constant Coefficients
Linear Nonhomogeneous Differential Equations Method of Undetermined Coefficients
Linear Homogeneous Differential Equations with Constant Coefficients Linear Nonhomogeneous Differential Equations
Ordinary Differential Equations
Linear Homogeneous Differential Equations with Constant Coefficients
Linear Nonhomogeneous Differential Equations Method of Undetermined Coefficients
Linear Homogeneous Differential Equations with Constant Coefficients Linear Nonhomogeneous Differential Equations
Exercises from last meeting
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
Linear Homogeneous Differential Equations with Constant Coefficients Linear Nonhomogeneous Differential Equations
Constructing a homogeneous equation from a specified solution
Linear Homogeneous Differential Equations with Constant Coefficients Linear Nonhomogeneous Differential Equations
Constructing a homogeneous equation from a specified solution Find a homogeneous linear equation with real, constant coefficients that is satisfied by : 1. y = 6 + 3xex − cos x
Linear Homogeneous Differential Equations with Constant Coefficients Linear Nonhomogeneous Differential Equations
Constructing a homogeneous equation from a specified solution Find a homogeneous linear equation with real, constant coefficients that is satisfied by : 1. y = 6 + 3xex − cos x 2. y = 4xex sin 2x 3. y = 100x2 − 26xex 4. y = x + e3x
Linear Homogeneous Differential Equations with Constant Coefficients Linear Nonhomogeneous Differential Equations
Ordinary Differential Equations
Linear Homogeneous Differential Equations with Constant Coefficients
Linear Nonhomogeneous Differential Equations Method of Undetermined Coefficients
Linear Homogeneous Differential Equations with Constant Coefficients Linear Nonhomogeneous Differential Equations
Method of Undetermined Coefficients Note that this method is only applicable when, and only when, the RHS of the equation is itself a particular solution of some homogeneous linear differential equation with constant coefficients. n
n−1
d y d y dy b0 (x) dx n + b1 (x) dxn−1 + · · · + bn−1 (x) dx + bn (x)y = R(x).
Linear Homogeneous Differential Equations with Constant Coefficients Linear Nonhomogeneous Differential Equations
Method of Undetermined Coefficients Note that this method is only applicable when, and only when, the RHS of the equation is itself a particular solution of some homogeneous linear differential equation with constant coefficients. n
n−1
d y d y dy b0 (x) dx n + b1 (x) dxn−1 + · · · + bn−1 (x) dx + bn (x)y = R(x).
Example Solve the equation y 00 + y 0 − 2y = 2x − 40 cos 2x.
Linear Homogeneous Differential Equations with Constant Coefficients Linear Nonhomogeneous Differential Equations
Method of Undetermined Coefficients Note that this method is only applicable when, and only when, the RHS of the equation is itself a particular solution of some homogeneous linear differential equation with constant coefficients. n
n−1
d y d y dy b0 (x) dx n + b1 (x) dxn−1 + · · · + bn−1 (x) dx + bn (x)y = R(x).
Example Solve the equation y 00 + y 0 − 2y = 2x − 40 cos 2x. → yc :
Linear Homogeneous Differential Equations with Constant Coefficients Linear Nonhomogeneous Differential Equations
Method of Undetermined Coefficients Note that this method is only applicable when, and only when, the RHS of the equation is itself a particular solution of some homogeneous linear differential equation with constant coefficients. n
n−1
d y d y dy b0 (x) dx n + b1 (x) dxn−1 + · · · + bn−1 (x) dx + bn (x)y = R(x).
Example Solve the equation y 00 + y 0 − 2y = 2x − 40 cos 2x. → yc : aux eqn : m2 + m − 2 = 0
Linear Homogeneous Differential Equations with Constant Coefficients Linear Nonhomogeneous Differential Equations
Method of Undetermined Coefficients Note that this method is only applicable when, and only when, the RHS of the equation is itself a particular solution of some homogeneous linear differential equation with constant coefficients. n
n−1
d y d y dy b0 (x) dx n + b1 (x) dxn−1 + · · · + bn−1 (x) dx + bn (x)y = R(x).
Example Solve the equation y 00 + y 0 − 2y = 2x − 40 cos 2x. → yc : aux eqn : m2 + m − 2 = 0 roots: m = −2, 1
Linear Homogeneous Differential Equations with Constant Coefficients Linear Nonhomogeneous Differential Equations
Method of Undetermined Coefficients Note that this method is only applicable when, and only when, the RHS of the equation is itself a particular solution of some homogeneous linear differential equation with constant coefficients. n
n−1
d y d y dy b0 (x) dx n + b1 (x) dxn−1 + · · · + bn−1 (x) dx + bn (x)y = R(x).
Example Solve the equation y 00 + y 0 − 2y = 2x − 40 cos 2x. → yc : aux eqn : m2 + m − 2 = 0 roots: m = −2, 1 yc = c1 e−2x + c2 ex
Linear Homogeneous Differential Equations with Constant Coefficients Linear Nonhomogeneous Differential Equations
Method of Undetermined Coefficients Note that this method is only applicable when, and only when, the RHS of the equation is itself a particular solution of some homogeneous linear differential equation with constant coefficients. n
n−1
d y d y dy b0 (x) dx n + b1 (x) dxn−1 + · · · + bn−1 (x) dx + bn (x)y = R(x).
Example Solve the equation y 00 + y 0 − 2y = 2x − 40 cos 2x. → yc : aux eqn : m2 + m − 2 = 0 roots: m = −2, 1 yc = c1 e−2x + c2 ex → yp :
Linear Homogeneous Differential Equations with Constant Coefficients Linear Nonhomogeneous Differential Equations
Method of Undetermined Coefficients Note that this method is only applicable when, and only when, the RHS of the equation is itself a particular solution of some homogeneous linear differential equation with constant coefficients. n
n−1
d y d y dy b0 (x) dx n + b1 (x) dxn−1 + · · · + bn−1 (x) dx + bn (x)y = R(x).
Example Solve the equation y 00 + y 0 − 2y = 2x − 40 cos 2x. → yc : aux eqn : m2 + m − 2 = 0 roots: m = −2, 1 yc = c1 e−2x + c2 ex → yp : R(x) is a solution to a homogeneous linear eqn whose aux eqn have roots mR = 0, 0, ±2i.
Linear Homogeneous Differential Equations with Constant Coefficients Linear Nonhomogeneous Differential Equations
Method of Undetermined Coefficients Note that this method is only applicable when, and only when, the RHS of the equation is itself a particular solution of some homogeneous linear differential equation with constant coefficients. n
n−1
d y d y dy b0 (x) dx n + b1 (x) dxn−1 + · · · + bn−1 (x) dx + bn (x)y = R(x).
Example Solve the equation y 00 + y 0 − 2y = 2x − 40 cos 2x. → yc : aux eqn : m2 + m − 2 = 0 roots: m = −2, 1 yc = c1 e−2x + c2 ex → yp : R(x) is a solution to a homogeneous linear eqn whose aux eqn have roots mR = 0, 0, ±2i. −2, 1, 0, 0, ±2i are roots of aux eqn of linear eqn with solution c1 e−2x + c2 ex + c3 + c4 x + c5 cos 2x + c6 sin 2x | {z } | {z } yc form of yp
Linear Homogeneous Differential Equations with Constant Coefficients Linear Nonhomogeneous Differential Equations
Solve the equation y 00 + y 0 − 2y = 2x − 40 cos 2x. yp = A + Bx + E cos 2x + F sin 2x where A, B, E, F are chosen such that yp is a particular solution to the given ODE.
Linear Homogeneous Differential Equations with Constant Coefficients Linear Nonhomogeneous Differential Equations
Solve the equation y 00 + y 0 − 2y = 2x − 40 cos 2x. yp = A + Bx + E cos 2x + F sin 2x where A, B, E, F are chosen such that yp is a particular solution to the given ODE. yp = − 12 − x + 6 cos 2x − 2 sin 2x
Linear Homogeneous Differential Equations with Constant Coefficients Linear Nonhomogeneous Differential Equations
Solve the equation y 00 + y 0 − 2y = 2x − 40 cos 2x. yp = A + Bx + E cos 2x + F sin 2x where A, B, E, F are chosen such that yp is a particular solution to the given ODE. yp = − 12 − x + 6 cos 2x − 2 sin 2x Hence, the general solution of the given ODE is y = c1 e−2x + c2 ex −
1 − x + 6 cos 2x − 2 sin 2x 2
Linear Homogeneous Differential Equations with Constant Coefficients Linear Nonhomogeneous Differential Equations
Exercises 1. y 00 − 5y 0 + 4y = 8ex 2. y 000 − y 00 = 3ex + sin x 3. y 000 + y 00 = ex cos x 4. y 00 − 8y 0 + 20y = 100x2 − 26xex
Linear Homogeneous Differential Equations with Constant Coefficients Linear Nonhomogeneous Differential Equations
Questions? See you next meeting!