What Ordinary Differential Equation (ODE)?

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Whatis ODE?

Ordinary differential equation (ODE) is an equation that involves a rate of change of a dependent variable with respect to one independent variable. The rate of change is denoted by one of these equal notations What Ordinary Differential Equation (ODE)? . Suppose we have a function What Ordinary Differential Equation (ODE)? , then variable What Ordinary Differential Equation (ODE)? is called domain or independent variable and variable What Ordinary Differential Equation (ODE)? is called range or dependent variable.

Because the rate change is involved only one variable, we call it ordinary derivative. This is in contrast to partial derivatives that involves rate of change of several variables. That is why we have ordinary differential equation (ODE) in contrast to partial differential equation (PDE).

Example:

What Ordinary Differential Equation (ODE)? is an ordinary differential equation with respect to one variable What Ordinary Differential Equation (ODE)? . The ODE is said to have 3-rd order because the highest order of the rate is 3. It is a first degree ODE because the power of the highest order is 1.

The following example illustrates the meaning of solution to differential equation.

Example:
Suppose we have a function What Ordinary Differential Equation (ODE)? . Differentiate this equation gives an ODE What Ordinary Differential Equation (ODE)? . Thus the general solution of ODE What Ordinary Differential Equation (ODE)? is What Ordinary Differential Equation (ODE)? . When we know the value of What Ordinary Differential Equation (ODE)? , we say that the particular solution of the ODE is What Ordinary Differential Equation (ODE)? . Notice that we reverse the strategy from solution to equation.

The general equation of the first order Ordinary Differential Equation (ODE) is given as

ODE

The right hand side is a function What Ordinary Differential Equation (ODE)? , which usually may involve the independent variable What Ordinary Differential Equation (ODE)? , the dependent variable What Ordinary Differential Equation (ODE)? and constants. Sometimes, either variable What Ordinary Differential Equation (ODE)? or What Ordinary Differential Equation (ODE)? are constant and it may simplify the case. In case the dependent variable What Ordinary Differential Equation (ODE)? is not presence, the right hand side becomes function What Ordinary Differential Equation (ODE)? and the solution of differential equation can be solved immediately by integration.

What Ordinary Differential Equation (ODE)? What Ordinary Differential Equation (ODE)? What Ordinary Differential Equation (ODE)? What Ordinary Differential Equation (ODE)?

Example below illustrates a case when both variable What Ordinary Differential Equation (ODE)? or What Ordinary Differential Equation (ODE)? are constants such that What Ordinary Differential Equation (ODE)? .

Example:
Suppose we found that population of our region has a constant rate of growth. If we put notation What Ordinary Differential Equation (ODE)? for the number of population in the region at time What Ordinary Differential Equation (ODE)? (measured in year), we can equate a constant growth rate with equation What Ordinary Differential Equation (ODE)? . Suppose we know that the growth rate is What Ordinary Differential Equation (ODE)? and this year population is 1200 people. What is the number of population 10 years from now? To answer that question we need to solve the ODE What Ordinary Differential Equation (ODE)? , which will give answer What Ordinary Differential Equation (ODE)? , inputting What Ordinary Differential Equation (ODE)? gives answer What Ordinary Differential Equation (ODE)?

When the dependent variable What Ordinary Differential Equation (ODE)? in the right hand side of general ODE equation What Ordinary Differential Equation (ODE)? is not a constant, the integration is not straightforward as above example. It needs differential algebraic equation as shown in the example below.

Example:

Solve What Ordinary Differential Equation (ODE)? for What Ordinary Differential Equation (ODE)? with initial value What Ordinary Differential Equation (ODE)?

As you may have learn in Calculus, we can write the equation as What Ordinary Differential Equation (ODE)? . We can separate What Ordinary Differential Equation (ODE)? and What Ordinary Differential Equation (ODE)? such that What Ordinary Differential Equation (ODE)? . Integrate both side of the equation

What Ordinary Differential Equation (ODE)? we get What Ordinary Differential Equation (ODE)? or What Ordinary Differential Equation (ODE)? , or What Ordinary Differential Equation (ODE)? . We can input initial condition What Ordinary Differential Equation (ODE)? into What Ordinary Differential Equation (ODE)? we get What Ordinary Differential Equation (ODE)? . Thus, the solution is What Ordinary Differential Equation (ODE)? . We plot for the range of x between 0 and 1

What Ordinary Differential Equation (ODE)?

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Teknomo, Kardi (2015) Solving Ordinary Differential Equation (ODE). http:\\people.revoledu.com\kardi\tutorial\ODE\