chapter2.page14

chapter2.page14 - a solution. For the linear rst order ODE...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
14 1. EXPLICITLY SOLVABLE FIRST ORDER DIFFERENTIAL EQUATIONS Here is how to get the graph in Mathcad : - we first defined a function x ( t,C ):= 0 . 2 0 . 02+ Ce - 0 . 2 t with two argu- ments t and C by typing x(t,C):0.2/0.02+C*e^-0.2t , - then we defined a range variable t:=0,0.1 ... 20 by typing t:0,0.1;20 , - finally, type @ at a blank area to get the xy-plot; in the function place holder, we put x(t,0),x(t,-0.01),x(t,-0.008),x(t,0.004),x(t,0.01) and type t in the variable place holder, and click at outside of the box to get the graph. a 4. Initial value problem and existence theorem The problem of finding a particular solution x ( t ) of an ODE x 0 = f ( t,x ) that satisfying condition x ( t 0 ) = x 0 is called an initial value problem. The problem can be written x 0 = f ( t,x ) x ( t 0 ) = x 0 Since solving an ODE requires a lots of hard work, it is wise to know if a given equation has any solution before march on the journey of find
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: a solution. For the linear rst order ODE x + a ( t ) x = b ( t ) we following existence-uniqueness theorem. Theorem 4.1 . If the function a ( t ) and b ( t ) are continuous on the open interval I containing the point t , then the initial value problem x + a ( t ) x = b ( t ) , x ( t ) = x has a unique solution x ( t ) on I, given by x ( t ) = e R a ( t ) dt Z b ( t ) e-R a ( t ) dt dt + C ! (8) with an appropriate value of C. So solving an initial value problem for linear rst order ODE is easy. One just apply the formula (8) and using the equation x ( t ) = x to nd C . Example 4.1 . Solve the initial value problem x +2 tx = 4 t 3 , x (0) = 1 . Solution First we apply the formula (8) x ( t ) = e R a ( t ) dt Z b ( t ) e-R a ( t ) dt dt + C ! ,...
View Full Document

Ask a homework question - tutors are online