Lecture 14: Ordinary Differential Equation I. First Order 1. Key points Maple commands dsolve 2. Introduction We consider a function of one variable. An ordinary differential equations (ODE) specifies the relation among its derivatives, the function itself and variable, e.g., where is a function. The order of the differential equations is determined by the highest order derivative in the equation. Equation (1) is a n-th order differential equation. Our goal is to determine that satisfies the differential equation. In general the solution contains n undetermined constants (constant of integration). We need additional conditions (boundary conditions) to determine the constants. In this lecture, we consider the first order ODE. A general form of first order differential equations can be written as However, the most common form in physics is Maple has extensive commands to solve ODEs. For simple case, you can use dsolve. See other tools in DETools. (1) (2) (3) 3. The simplest case: y'=f(x) This kind of ODE can be converted to a simple integral.
This is a special case of y'=u(x)/w(y) discussed in section 5. Example: time-dependent force Consider a particle of mass m subject to an external forcing given by. Its equation of motion is What is the velocity at a later time t? (4) where is an initial velocity. Its position can be obtained by integrating the velocity. (5) Using Maple, (6) where _C1 is a constant of integration, equivalent to in Eq (4). You can specify boundary conditions as a part of the definition of ODE. For example, if the particle was initially at rest (7)
1 8 6 4 2 2 4 6 8 1 t 4. Autonomous equation: y'=f(y) If does not depend on, we have an autonomous equation equation to an integral.. We can convert the differential At certain values of, may become zero. Then, the integral may diverge. However, as you see below, cannot pass through those points since at those points. When, is positive. Hence, increases as increases. When, y' is negative. Hence, decreases as increases. When,. That means does not change. remains at the root of
From this flow diagram, it is clear that the 1st order autonomous ODE has only two types of solutions: one monotonically approaches to a fixed point (attractor) and the other monotonically diverges. Example: Free falling Consider a particle of mass m subject to a frictional force is falling under a uniform gravity g. Assume that its velocity is at. The equation of motion for the particle is and its integral solution is (8) (9) Solving for, we obtain (1)
Since (terminal velocity) is only the attractor in this equation, all solutions monotonically approach to the terminal velocity. (11) where _C1 is a constant of integration. To specify a boundary condition, (12) in agreement with Eq. (1). You can plot solutions as a flow diagram using DEplot 5 2 4 6 8 1 t v(t) Regardless of the initial value of, all the velocities approach the terminal velocity. 5. Separable variables: y'=u(x)/w(y)
When an ODE can be written in a form,, we say and are separable. This ODE can be also converted to integrals. Note that the previous forms of ODEs are special cases of this type. Example 1: Boyle's gas law Boyle's gas law states (13) where V and P are volume and pressure of the gas. Separating the variables, Integrating it, we have a general solution with a constant of integration: (14) Combining two log functions, we obtain a popular expression of Boyle's law where k is a constant. (15) (16) Maple solves it in one step: (17) Example 2: Overdumped parametric oscillator Consider the motion of spring with time-varying spring constant. For example, where is a friction constant and is a oscillating spring constant. If the friction is very large, we can ignore the inertial mass (over dumped limit), then the equation of motion becomes (18)
(19) Using the method of separation of variables, we have the both sides, we obtain. Integrating where is the position at time and is an initial position. Combining the log functions, we have. Therefore, the position of the particle is found to Maple solve it (2) (21) where is the integral constant determined by an initial condition. No matter how strong the oscillation is, the particle always relaxes to the equilibrium position as shown below.
1 8 6 X(T) 4 2 2 4 6 8 1 T 6. More exercises Nonlinear friction When a particle moves in a fluid, it experiences a drag force where is the velocity of the particle and. Then, the equation of the motion is given by. Find the time evolution of the velocity. solution For, (22)
Solving (22) for V, we obtain Warning, solve may be ignoring assumptions on the input variables. (23) In particular for The velocity evolves as (24) While (23) goes to when. It goes to zero much slower than the exponential relaxation for the linear friction case. The solution (23) is not valid for the linear friction case ( ). This linear friction is already investigated in the section 4. Using Maple command, (25) which is the same as (23). Separable ODE Solve the ODE:. solution Using the variable separation method, the ODE can be expressed as. Integrating it, we obtain Rearranging it, we have. Hence,. 7. Homework
Solve the differential equation and plot the flow diagram and two solution curves corresponding to initial conditions (a) and (b).