This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Linear First Order Differential Equations These equations take the general form dy dx = p ( x ) y + g ( x ) , i . e ., dy dx + p ( x ) y = g ( x ) , where p ( x ) and g ( x ) are continuous (piecewise continuous is OK) func tions on some interval a x b . We refer to these first order differ ential equations as linear differential equations because the unknown y appears linearly, i.e., to the first power, with a known coefficient, p ( x ). The equation is homogeneous if g ( x ) 0, for given functions g ( x ) other than zero it is said to be inhomogeneous . Homogeneous Case: Equation: dy dx + p ( x ) y = 0. Properties: y ( x ) 0 is a solution; the trivial solution. (Clear.) If y ( x ) is any solution of the homogeneous equation then, for any constant c , the multiple c y ( x ) is also a solution of that equation. (Just substitute c y ( x ) into the equation.) If a solution y ( x ) is nonzero for some value, say x 1 , of x , then it is nonzero for all values of x . (Will be shown.) Method of Solution: In the homogenous case we have dy dx = p ( x ) y . Assuming that we are looking for a nonzero solution, we can write 1 y ( x ) dy dx = p ( x ) . Integrating both sides with respect to x , we have (natural logarithm) log  y ( x )  = Z x p ( s ) ds + c  P ( x ) + c, 1 where c is an arbitrary constant and P ( x ) is an antiderivative, or in definite integral, of p ( x ). Taking the exponential of both sides we have the general solution...
View
Full
Document
This note was uploaded on 01/23/2012 for the course MATH 4254 taught by Professor Robinson during the Spring '10 term at Virginia Tech.
 Spring '10
 ROBINSON
 Equations

Click to edit the document details