Differential equations lecture no.25. First and second order differential equations

POINT 1. GENERAL NOTIONS

POINT 2. INTEGRABLE TYPES OF THE FIRST ORDER DIFFERENTIAL EQUATIONS

POINT 3. ORDER REDUCING SECOND ORDER DIFFERENTIAL EQUATIONS

Point 1. General notions

Def. 1. Equation with respect to an unknown function is called differential one if it contains the derivative or derivatives of this function.

Def. 2. Order of a differential equation is called the highest order of derivatives of the unknown function which contains the equation.

( 1 )

is the general form of the first order differential equation in .

( 2 )

is the general form of the second order differential equation in .

Def. 3. Solution of a differential equation is called a function which satisfies this equation that is turns it into identity.

For example a function is the solution of the first order differential equation (1) if .

Def. 4. Integral curve of a differential equation is called the graph of its solution.

Every differential equation has infinitely many solutions.

Ex. 1. Set of all solutions of a differential equation is given by the expression where C is an arbitrary constant.

To choose some certain solution of a differential equation one gives additional conditions.

There are boundary and initial conditions.

Ex. 2. Find a solution of a differential equation

which satisfies the next boundary conditions

.

In our lectures we’ll study as the rule differential equations with initial conditions.

For the first order differential equation (1) we have one initial condition name-ly

. ( 3 )

For the second order differential equation (2) we have two initial conditions

. ( 4 )

Def. 5. Problem of finding solutions of a differential equation which satisfy an initial condition or initial conditions is called the initial-value problem or Cauchy problem.

For the first order differential equation (1) we have Cauchy problem (1), (3), and for the second order differential equation (2) we have Cauchy problem (2), (4).

Geometrical sense of Cauchy problem (1), (3): find integral curves which pass through a given point .

Geometrical sense of Cauchy problem (2), (4): find integral curves which pass through a given point and have at this point the given slope of a tangent.

Ex. 3. Find a curve through a point if the slope of the tangent at its arbitrary point equals .

We must solve Cauchy problem for the first order differential equation

with an initial condition

.

The equation gives

,

and from the initial condition we find the value of C,

,

and therefore

.

Theory of differential equations establishes conditions for one-valued solvabi-lity of Cauchy problems.

Let’s we study the first order differential equation which is resolved with respect to the derivative that is

( 5 )

Theorem 1. If a function in the first order differential equation (5) and its partial derivative with respect to y are continuous in some domain D of the -plane, then for any point Cauchy problem (5), (3) has unique solution.

Def. 6. General solution of the first order differential equation (5) in the domain D of the theorem 1 is called a function which contains an arbitrary constant C and satisfies two conditions: a) it is a solution of the equation for any va-lue of C; b) for any initial condition (3) one can find a value of C such that the function satisfies this condition.

Def. 7. If an arbitrary constant C in the general solution of a differential equation (5) takes on some particular value then the corresponding function is called a particular solution of this equation.

For example the solution of Cauchy problem is a particular one.

Let’s now we study the second order differential equation resolved with respect to the second derivative

( 6 )

Theorem 2. If a function in the differential equation (6) and its partial derivatives with respect to are continuous in some domain D of the -space, then for any point Cauchy problem (6), (4) has unique solution.

Def. 8. General solution of the second order differential equation (6) in the do-main D of the theorem 2 is called a function which contains two arbitrary constants and satisfies two conditions: a) it is a solution of the equation for any values of ; b) for any initial conditions (4) one can find values , of such that the function satisfies these conditions.

For any particular values , of we have a particular solution y = of the equation (6), for example the solution of Cauchy problem (6), (4).

Note 1. The general solution or a particular solution of a differential equation can be expressed implicitly. In such case they can be called respectively the general integral or a particular integral of this equation.

Note 2. Analogous definitions and theorems are introduced for nth order differential equations (if n > 2).

In the future the expression “to solve (or to integrate) a differential equation” means: a) to find its general solution or b) to solve Cauchy problem for the equation.