Let f be a continuous real host function of a real variable. Show that an autonomous differential equation dy / dt=f(y) can only have montont growing, montont decreasing or constant solutions. (In particular, they cannot therefore have oscillatory solutions)<br /><br />please don't comment here Agar ni ata hoga to
Answers
Answer:
An autonomous differential equation is an equation of the form
dydt=f(y).
Let's think of t as indicating time. This equation says that the rate of change dy/dt of the function y(t) is given by a some rule. The rule says that if the current value is y, then the rate of change is f(y).
The equation is called a differential equation, because it is an equation involving the derivative dy/dt. The differential equation is called autonomous because the rule doesn't care what time t it is. It only cares about the current value of the variable y.
Given an autonomous differential equation, we'll often want to solve the equation, which means find a function a y(t) whose derivative dy/dt is equal to f(y).
A linear differential equation
An example of an autonomous differential equation, let's let f(y) be the linear function f(y)=2y, so the equation is
dydt=2y.
which we could also write as
dydt(t)=2y(t),
but, usually, we won't write out the explicit dependence on t.
We've ended up with a linear differential equation, one of the simplest types.
Solution methods
How can we solve this equation, i.e., find a function y(t) that, if we differentiate, we get the function y(t) back, only multiplied by two? We have three main methods for solving autonomous differential equations.
Numerical methods. We can approximate the continuous change of the differential equation with discrete jumps in time, By doing this, we get a formula for evolving from one time step to the next (like a a discrete dynamical system). We can then use a computer to calculate this approximation to the solution.
Graphical methods. We can use a plot of the graph of f(y) to determine the behavior of y(t).
Analytic methods. We can use mathematics to find a function y(t) that satisfies the differential equation.
The Guess and Check method
Here, let's try an analytic method. We'll use an important analytic method, which is called guess and check. Guess and check is a perfectly valid method because, if you can find any function that satisfies the equation, you've found a solution. It doesn't matter what method you use to find the function; as long as you can show it is a solution, you've accomplished the task.
The way to guess is to use your knowledge and intuition above derivatives to find a function whose derivative behaves in the required way.
Here we want to find a function where, if we take the derivative, we get the function back, only multiplied by 2. Let's forget about the factor of two for a moment. Do you know any function that is it's own derivative?
The exponential function fits the bill. Remember that
ddtet=et.
so if y(t)=et, then dydt=et=y(t).