K1 SOLUTIONS TO DIFFERENTIAL EQUATIONS K3A FIRSTORDER ORDINARY

STAT 103 HOMEWORK THREE SOLUTIONS SPRING 2014 INSTRUCTIONS THE
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Derive Lecture 9&10, page 2/8


K.1 Solutions to Differential Equations

K.3.A First-order Ordinary Differential Equations

Also see http://www.ucl.ac.uk/Mathematics/geomath/level2/deqn/ de8.html

K1 SOLUTIONS TO DIFFERENTIAL EQUATIONS  K3A FIRSTORDER ORDINARY (K-15)

Integrating factor = exp K1 SOLUTIONS TO DIFFERENTIAL EQUATIONS  K3A FIRSTORDER ORDINARY

Multiply through the integrating

K1 SOLUTIONS TO DIFFERENTIAL EQUATIONS  K3A FIRSTORDER ORDINARY

Collect term

K1 SOLUTIONS TO DIFFERENTIAL EQUATIONS  K3A FIRSTORDER ORDINARY

K1 SOLUTIONS TO DIFFERENTIAL EQUATIONS  K3A FIRSTORDER ORDINARY

and divide by K1 SOLUTIONS TO DIFFERENTIAL EQUATIONS  K3A FIRSTORDER ORDINARY

K1 SOLUTIONS TO DIFFERENTIAL EQUATIONS  K3A FIRSTORDER ORDINARY (K-16)

Example A–1 Integrating Factor for Series Reactions


K1 SOLUTIONS TO DIFFERENTIAL EQUATIONS  K3A FIRSTORDER ORDINARY

Integration factor K1 SOLUTIONS TO DIFFERENTIAL EQUATIONS  K3A FIRSTORDER ORDINARY

K1 SOLUTIONS TO DIFFERENTIAL EQUATIONS  K3A FIRSTORDER ORDINARY

K1 SOLUTIONS TO DIFFERENTIAL EQUATIONS  K3A FIRSTORDER ORDINARY

K1 SOLUTIONS TO DIFFERENTIAL EQUATIONS  K3A FIRSTORDER ORDINARY

K1 SOLUTIONS TO DIFFERENTIAL EQUATIONS  K3A FIRSTORDER ORDINARY

K1 SOLUTIONS TO DIFFERENTIAL EQUATIONS  K3A FIRSTORDER ORDINARY

K.3.B Coupled First-order Linear Ordinary Differential Equations with Constant Coefficients

Also see http://www.mathsci.appstate.edu/~sjg/class/2240/finalss04/ Alicia.html.

Consider the following coupled set of linear first order ODE with constant coefficients.

Or in Matrix Notation

K1 SOLUTIONS TO DIFFERENTIAL EQUATIONS  K3A FIRSTORDER ORDINARY


(1) K1 SOLUTIONS TO DIFFERENTIAL EQUATIONS  K3A FIRSTORDER ORDINARY K1 SOLUTIONS TO DIFFERENTIAL EQUATIONS  K3A FIRSTORDER ORDINARY K1 SOLUTIONS TO DIFFERENTIAL EQUATIONS  K3A FIRSTORDER ORDINARY

(2) K1 SOLUTIONS TO DIFFERENTIAL EQUATIONS  K3A FIRSTORDER ORDINARY K1 SOLUTIONS TO DIFFERENTIAL EQUATIONS  K3A FIRSTORDER ORDINARY K1 SOLUTIONS TO DIFFERENTIAL EQUATIONS  K3A FIRSTORDER ORDINARY


The solution to these coupled equations is

K1 SOLUTIONS TO DIFFERENTIAL EQUATIONS  K3A FIRSTORDER ORDINARY (3)

and

K1 SOLUTIONS TO DIFFERENTIAL EQUATIONS  K3A FIRSTORDER ORDINARY (4)

where

K1 SOLUTIONS TO DIFFERENTIAL EQUATIONS  K3A FIRSTORDER ORDINARY

From the initial conditions

t = 0 x = x0 and y = y0 we get

K1 SOLUTIONS TO DIFFERENTIAL EQUATIONS  K3A FIRSTORDER ORDINARY

and

K1 SOLUTIONS TO DIFFERENTIAL EQUATIONS  K3A FIRSTORDER ORDINARY

differentiating equations (3) and (4) and evaluating the derivative at t = 0

K1 SOLUTIONS TO DIFFERENTIAL EQUATIONS  K3A FIRSTORDER ORDINARY

K1 SOLUTIONS TO DIFFERENTIAL EQUATIONS  K3A FIRSTORDER ORDINARY

We have four equations and four unknowns (The arbitrary constants of integration A, B, K1 and K2) so we can eliminate these arbitrary constants of integration.

If y0 = 0 then the solution takes the form

K1 SOLUTIONS TO DIFFERENTIAL EQUATIONS  K3A FIRSTORDER ORDINARY

K1 SOLUTIONS TO DIFFERENTIAL EQUATIONS  K3A FIRSTORDER ORDINARY

3

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Tags: differential equations, ordinary differential, differential, ordinary, solutions, equations, firstorder