Differential Equations

Differential Equations 1. Definition of Differential Equations A differential equation is an equation expressing a relation between a function and its derivatives. For example, we might know that x is a func­ tion of t and .. . x + 8x + 7x = 0. (1) or perhaps the relation is more complicated, like � xx (5) + cos(t)etx + ( x �� x � x )6 = sin(5t). (2) When the function in the differential equation has a single independent variable we call it an ordinary differential equation. That is, the deriva­ tives are ordinary derivatives, not partial derivatives. This course is almost exclusively concerned with ordinary differential equations. The Order of a Differential Equation The order of a differential equation is the order of the largest derivative ap­ pearing in it. Equation (1) is a second order differential equation. Equation (2) is a fifth order equation since the highest derivative is x (5) (in the first term). 2. Solving a Differential Equation Solving a differential equation means finding a function that satisfies the equation. For many equations it can be hard or impossible to find a solution. One thing that is easy however is to check a proposed solution. We demonstrate with a few examples. Example 1. Checking a Solution By Substitution Verify the y(t) = e3t is a solution to the differential equation . y = 3y. (3) Solution. To do this we simply substitute y = e3t into (3), and check that . the equation holds. On the left hand side of (3) we have y = 3e3t . On the right hand side we have 3y = 3e3t . Since both sides are equal, y = e3t is a solution. Example 2. Rejecting a Solution by Substitution Show that y(t) = t3 is not a solution to the differential equation . y = y/t. (4) OCW 18.03SC Solution. Again, we substitute the expression for y into (4). . Left hand side: y = 3t2 . Right hand side: y/t = t2 . Since the two sides are not equal, y = t3 is not a solution. 3. Parametrizing the Set of Solutions of a Differential Equation Differential equations usually have more than one solution. We can describe them all at once using a parameter. Example. Find all the solutions to .. x = 2t (5) This is a standard calculus problem. Integrating twice and remembering to include the constants of integration gives x (t) = t3 + c1 t + c2 , 3 where c1 and c2 are arbitrary constants. This expression gives a parametriza­ tion of the set of solutions to equation (5). The constants c1 and c2 are param­ eters. Every choice of c1 and c2 gives a different solution to (5). For example, x = t3 /3 + 2t + 1 and t3 /3 + πt + 2.718 are both solutions. 4. Initial Value Problems Sometimes we have a differential equation and initial conditions. To­ gether they make up an initial value problem. The meaning of the term initial conditions is best illustrated by example. .. Example. Solve the initial value problem x = 2t with the initial conditions . x (1) = 1, x (1) = 2. Solution. In the previous example we found the general solution of this differential equation t3 x ( t ) = + c1 t + c2 . 3 We use the initial conditions to find the values of c1 and c2 . . . x (t) = t2 + c1 ⇒ x (1) = 1 + c1 = 2. x (t) = t3 /3 + c1 t + c2 ⇒ x (1) = 1/3 + c1 + c2 = 1. Solving for c1 and c2 we get c1 = 1, c2 = −1/3. Thus, the solution to the initial value problem is x (t) = t3 /3 + t − 1/3. 18.03SC 5. Acronyms It will be convenient at times to allow ourselves to use acronyms. Some of the most common are 1. Differential equation (DE). 2. Ordinary differential equation (ODE). 3. Initial value problem (IVP). 4. Initial conditions (IC).