Ch 2.4: Differences Between Linear and Nonlinear Equations. Recall that a first order ODE has the form y' = f ( t , y ), and is linear if f is linear in y, and nonlinear if f is nonlinear in y. Examples: y' = t y - e t , y' = t y 2 .
(Example) Find the solution of the Initial Value Problem (IVP). How many solutions does IVP have?
when does IVP have the unique (only one) solution?
(t, y) (, ) x (, )containing the point (t0, y0).
Then in some interval (t0 - h, t0 + h) (, ) there exists a unique solution
y = (t)that satisfies the IVP.
t > 0, the interval on which p(t) = 2/t is continuous: t > 0 or t < 0
solution is unique on corresponding interval.
and are continuous except on line y = 1.
which are continuous except on line y = 1.
(Question) What is the interval of definition of the solution?
exists throughout any interval about t = t0 on which p and g are continuous.
*** From the coefficients we can predict the solution’s shape and behavior.
exists throughout any interval about t = t0 on which p and g are continuous, and this solution is unique.
and can be evaluated at any appropriate value of t, as long as the necessary integrals can be computed.
chapter and consider the following example
A ball with mass 0.1 (Kg) is thrown upward with initial velocity 30 m/sec from the roof of a building 40 m high (g = 9.8 ). Neglect air resistance.
( a) Set up an ODE with initial condition and find the solution to find the velocity of the ball in time t (sec).
(b) Find the maximum height above the ground that the ball reaches.