Ma1506 Leture Notes
MA1506 LECTURE NOTES CHAPTER 1 DIFFERENTIAL EQUATIONS 1.1 Introduction
A differential equation is an equation that contains one or more derivatives of a differentiable function. [In this chapter we deal only with ordinary DEs, NOT partial DEs.] The order of a d.e. is the order of the equation’s highest order derivative; and a d.e. is linear if it can be put in the form
any (n)(x)+an−1y (n−1)(x)+· · ·+a1y (1)(x)+a0y(x) = F,
1
where ai, 0 ≤ i ≤ n, and F are all functions of x. For example, y = 5y and xy − sin x = 0 are first order linear d.e.; (y )2 + (y )5 − y = ex is third order, nonlinear. We observe that in general, a d.e. has many solutions, e.g. y = sin x + c, c an arbitrary constant, is a solution of y = cos x. Such solutions containing arbitrary constants are called general solution of a given d.e.. Any solution obtained from the general solution by giving specific values to the arbitrary constants is called a particular solution of that d.e. e.g.
2
y = sin x + 1 is a particular solution of y = cos x. Basically, differential equations are solved using integration, and it is clear that there will be as many integrations as the order of the DE. Therefore, THE GENERAL SOLUTION OF AN nth-ORDER DE WILL HAVE n ARBITRARY CONSTANTS.
1.2
Separable equations
A first order d.e. is separable if it can be written in the form M (x) − N (y)y = 0 or equivalently, M (x)dx = N (y)dy. When we write the
3
d.e. in this form, we say that we have separated the variables, because everything involving x is on one side, and everything involving y is on the other. We can solve such a d.e. by integrating w.r.t. x:
M (x)dx = N (y)dy + c. Example 1. Solve y = (1 + y 2)ex.
Solution. We separate the variables to obtain 1 e dx = dy. 1 + y2 x 4
Integrating w.r.t. x gives ex = tan−1 y + c, or tan−1 y = ex − c, or y = tan(ex − c). Example 2. Experiments show that a ra-
dioactive substance decomposes at a rate proportional to the amount present.
A differential equation is an equation that contains one or more derivatives of a differentiable function. [In this chapter we deal only with ordinary DEs, NOT partial DEs.] The order of a d.e. is the order of the equation’s highest order derivative; and a d.e. is linear if it can be put in the form
any (n)(x)+an−1y (n−1)(x)+· · ·+a1y (1)(x)+a0y(x) = F,
1
where ai, 0 ≤ i ≤ n, and F are all functions of x. For example, y = 5y and xy − sin x = 0 are first order linear d.e.; (y )2 + (y )5 − y = ex is third order, nonlinear. We observe that in general, a d.e. has many solutions, e.g. y = sin x + c, c an arbitrary constant, is a solution of y = cos x. Such solutions containing arbitrary constants are called general solution of a given d.e.. Any solution obtained from the general solution by giving specific values to the arbitrary constants is called a particular solution of that d.e. e.g.
2
y = sin x + 1 is a particular solution of y = cos x. Basically, differential equations are solved using integration, and it is clear that there will be as many integrations as the order of the DE. Therefore, THE GENERAL SOLUTION OF AN nth-ORDER DE WILL HAVE n ARBITRARY CONSTANTS.
1.2
Separable equations
A first order d.e. is separable if it can be written in the form M (x) − N (y)y = 0 or equivalently, M (x)dx = N (y)dy. When we write the
3
d.e. in this form, we say that we have separated the variables, because everything involving x is on one side, and everything involving y is on the other. We can solve such a d.e. by integrating w.r.t. x:
M (x)dx = N (y)dy + c. Example 1. Solve y = (1 + y 2)ex.
Solution. We separate the variables to obtain 1 e dx = dy. 1 + y2 x 4
Integrating w.r.t. x gives ex = tan−1 y + c, or tan−1 y = ex − c, or y = tan(ex − c). Example 2. Experiments show that a ra-
dioactive substance decomposes at a rate proportional to the amount present.