University Mathematics Handbook (2015)

XI. Ordinary Differential Equations, or ODE

Chapter 1. Classification of First-Order Ordinary Differential Equations

1.1  Introduction

a. An ordinary differential equation, or ODE, expresses the relation between a one-variable function and its derivatives.

The general form of -th order ODE for function is

when is a given function of variables.

b. Function is a solution of n-th-order ODE on interval if it is -times differentiable and satisfies the equation

c. A first-order ODE is the equation or, in its explicit form, is .

d. Example: The simplest ODE is , and its solutions are when is a constant.

e. A general solution of a first-order ODE includes all solutions and is dependent on a constant . Every specific choice of , results in a particular solution.

The particular solution of that satisfies is .

f. The problem consisting of equation and initial conditions is called a Cauchy problem.

1.2  Separable Equations

a.  Equations , have separate variables. Such an equation is called separable equation.

b.  A general solution of is

c.  A general solution of the equation is

Division of both sides by usually loses particular solutions holding .

d. The equation is not separable, but turns into separable equation after substituting and its solutions are

.

1.3  Homogeneous Equations

a. is a -th-order homogeneous function if for every real .

Example: Function where all the powers of the numerator and of the denominator are equal is a -order homogeneous function.

b. ODE is homogeneous if is -order homogeneous function.

c. Substituting transforms the homogeneous equation into the separable equation .

d. An equation in the form of turns into homogeneous equation if the coordinates are shifted to point , the intersection of lines

.

After substituting , , we obtain homogeneous equation

1.4  Exact Equations

a. The equation when in domain is called exact equation. In this case there exist continuous function with continuous partial derivatives, such that and the general solution of the exact equation is .

b. If , then

or

1.5  Integrating Factor

a. If the equation is not exact, but, if multiplied it by function , the result is the equation

which is an exact equation, that is

Then, function is called an integrating factor.

b. If the expression is a function of only, then the integrating factor is a function dependent of only, and is the solution of the equation

c. If the expression is a function of only, then the integrating factor of the equation is dependent of only, and is the solution of the equation

d. If the expression is a function of only, then the integrating factor is a function dependent of , and is the solution of the equation when .

1.6  First-Order Linear Equations

a. The ODE

(*)

when and are continuous functions on is called a first-order linear equations.

b. If , the equation is a homogeneous linear equation.

c. General solution of homogeneous linear equation is .

d. is a general solution of (*).

e.  Existence and Uniqueness Theorem: If functions and are continuous in open interval , then there exists a unique function holding (*) and for every given .

f.  Bernoulli's Equation: . Substituting , we get a linear equation

g.  Riccati Equation: .

If one particular solution of equation is known, then by substituting we get a Bernouli equation for .

1.7  Existence and Uniqueness Theorem for First-Order ODE

Given an ODE with initial conditions (Cauchy's problem):

, (*)

a.  The integral form of (*):

b.  Existence and Uniqueness Theorem: if function is continuous on rectangle

and satisfy Lipschitz continuity criteria in

when is constant and on , then (*) has a unique solution on when .

c.  If and are continuous on and in rectangle , then Lipschitz continuity criteria holds.

d.  The unique solution of a Cauchy's problem can be found using the Picard iterations, based on constructing a sequences of functions following the formula

which converges to a solution.

e.  If, under the existence and uniqueness in b, does not hold the Lipschitz continuity criteria, then (*) has a least one solution.

f.  If is continuous and has continuous partial derivatives to -order, included, in the neighborhood of point , then the solution of Cauchy's problem (*) is an times differentiable function.