Linear Algebra in Differential Equations (1) (1)

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Linear Algebra in Differential Equations:

Linear Algebra in Differential Equations Josal Patel and Bharat Venkat

What is a Differential Equation?:

What is a Differential Equation? A differential equation is an equation consisting of one or more derivatives of an unknown function. Examples: y’ − 2y = 0 Nonexamples: y + 3 = 49 cos(3x+ 7y) + 57 = 465

Why do we need Linear Algebra?:

Why do we need Linear Algebra? Linear algebra has significant applications in differential equations Differential equations are used as models for many of the laws that govern the natural world. An example of this is Newton’s second law which is expressed as a second-order differential equation. Without this application, it would be much harder to solve and manipulate differential equations and use them as models for systems in physics, applied mathematics, biology, economics, etc … Linear algebra techniques used include: Use of matrices to express systems of differential equations. Finding the eigenvalues and eigenvectors of matrices to solve systems of differential equations .

Why do we need Linear Algebra?:

Why do we need Linear Algebra? Systems of Differential Equations have multiple equations with multiple variables Need a way to combine information from multiple equations. Some ODE’s are difficult to solve, linear algebra can simplify process Variation of Parameters and other differential equation techniques are integral heavy, sometimes very complicated. How does Linear Algebra help? Express complicated differential equations as systems of less complicated equations. Use eigenvectors and eigenvalues to solve equations instead of taking multiple integrals. Differentiation is a linear operator.

Classification of Differential Equations:

Classification of Differential Equations Order Highest derivative in equation Linearity Linear if variables and derivatives are only multiplied by constants Ex. y’’ + y’ + 7 = 8 is a 2nd order linear differential equation

Homogeneous or not?:

Homogeneous or not? Homogeneous differential equation equals 0 Ex. y’ + 9y = 0 Non-homogeneous differential equation does not equal 0 Ex. y”” + 3y = 3cos(x+3)

Describing Differential Equations as a System:

Describing Differential Equations as a System Goal is to describe higher order differential equation as a system of first order linear differential equations Use change of variables Differentiate both sides

Describing Differential Equations as a System:

Describing Differential Equations as a System Goal is to describe higher order differential equation as a system of first order linear differential equations Example: Given differential equation 2y’’-5y’ + y = 0 Use change of variables x 1 = y and x 2 = y’ Differentiate: x 1 ’ = y’ = x 2 x 2 ’ = y’’ = (5/2)y’ -y/2 = (5/2)x 2 - x 1 /2

Matrix Describing System of Differential Equations:

Matrix Describing System of Differential Equations Given a system of differential equation s to put it into matrix form: Express each side as a vector Express the right side as matrix multiplication of a coefficient matrix and variables

Matrix Describing System of Differential Equations:

Matrix Describing System of Differential Equations Example: Convert x 1 ’ = 4x 1 + 7x 2 into matrix form x 2 ‘ = -2x 1 - 5x 2

Matrix Describing System of Differential Equations:

Matrix Describing System of Differential Equations Matrix equation for system x’ = ax

Solving Systems of Differential Equations:

Solving Systems of Differential Equations Once the system of differential equations has been converted to matrix form, you must find the eigenvalues and eigenvectors of the coefficient matrix. The general solution to the system can be expressed as: where v is the eigenvector and λ   is the eigenvalue of matrix A

Solving Systems of Differential Equations:

Solving Systems of Differential Equations Example: y’ 1 = y 1 + 2y 2 y’ 2 = -y 2

Solving Systems of Differential Equations:

Solving Systems of Differential Equations Example ( cont …): y’ 1 = y 1 + 2y 2 y’ 2 = -y 2 Then find the corresponding eigenvectors: V = V =

Solving Systems of Differential Equations:

Solving Systems of Differential Equations Example ( cont …): y’ 1 = y 1 + 2y 2 y’ 2 = -y 2 Hence, the solution is given by:

Sources :

Sources Elementary Differential Equations by Jeffrey Ledford http:// tutorial.math.lamar.edu /Classes/DE/ SystemsDE.aspx

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