LECTURE Differential eq


LECTURE 13, 14

Differential Equations

First Order Equations with Separable Variables (separable equations), First Order Homogeneous Linear Equations, First Order Linear Equations- method of variation of constants, undetermined coefficients, Homogeneous Equations, Second order differential equations

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FIRST ORDER DIFFERENTIAL EQUATION

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y(t0) = y0 - initial value

Definition

The general solution of a first order differential equation is the family of all functions y(t) that satisfy the equation. Each function in this family is called a particular solution of this equation.

In most cases, the family of functions will depend in some way on a constant C, and the graphs of these functions will form a family of curves that fill up the (t, y) plane, but do not touch each other, as in the Figure 1.

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Figure 1

Example:

The general solution of

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See the next Figure:

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FIRST ORDER DIFFERENTIAL EQUATION WITH SEPARABLE VARIABLES

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The graph of the solution:

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The above equation has separable variables, because it can be written

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The particular solution is x(t) = C t -3 , take y(x) = C(t)t-3

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General solution is

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The required particular solution is

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UNDETERMINED COEFFICIENTS

The method of undetermined coefficients of solving a nonhomogeneous equation

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is based on

Theorem

The general solution of a linear nonhomogeneous differential equation is the sum of a particular solution and the general solution of the homogeneous equation

The general solution of the homogeneous equation is determined as above and the particular solution is guessed.

The guess of the particular solution is relatively easy if :

the function p (x) in the equation is constant (it is then a linear differential equation with constant coefficients), and the function f (x) is:

In each of these cases a particular integral of the nonhomogeneous equation should be guessed in the same form as f(x), conserving the degree of a polynomial, as well as numbers   and b.

Remaining constants (coefficients of the polynomial, α, β and a) are left undetermined. Their values can be found by back substituting the guess into the nonhomogeneous equation

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It must be noted that this method, unlike the method of variation of coefficients, is not general. However, in  the above cases it is computationally simpler.

Example

Using the method of undetermined coefficients we will find a complete integral of the equation

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Here p (x)  4 (a constant function), and f (x) = x3 (a polynomial).

1. Find the general solution

The general solution is of the form

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2. Find a particular solution using the method of undetermined coefficients

Since

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we guess that the particular solution is a polynomial of the 3rd degree, i.e.,

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hence

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Substituting back into the equation we get

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that is

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This equality is satisfied for every x if and only if the following system of equations is satisfied

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Hence

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Thus the function

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is the desired particular solution, which implies that

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is the general solution of the equation

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.

Example

Use the method of undetermined coefficients to the find the particular solution of the equation

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with the initial condition y(0) = 2.

It is a nonhomogeneous linear equation of the first order with constant coefficients (p(x)  1), where f(x) = xex .

1. We determine the general solution. It is of the form:

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2. We determine the particular solution using the method of undetermined coefficients

We assume it is a product of the polynomial of the first order and the exponential function ex

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Hence

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Thus

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This condition is satisfied if and only if

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that is, for any x

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Hence

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which gives

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Thus the particular solution is of the form

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whereas the general solution is

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Taking into account the initial condition y(0) = 2, we obtain

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The desired particular solution is

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HOMOGENEOUS EQUATIONS

Let f be a function continuous on an interval (a, b), such that f (u) ≠ u on (a,b).

Definition

A homogeneous differential equation is an equation that can be written in the form

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Equations of this type can be reduced to a separable equation by the substitution

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Indeed, differentiating the equality

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we get

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Substituting in the equation

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Separation of variables yields

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To return to the initial function y ,we substitute u = yx,

Remark

If the condition f (u) ≠ u is not satisfied on (a,b), consider the equation f (u) = u.

Example

We will find the solution to the equation

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Substituting

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we obtain

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and further

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For f (u)u, that is, if 0x01 graphic
we separate variables and integrate

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That is

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For f (u) = u we have

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Hence the additional solution y ≡ 0.

Quite often, in order to identify an equation to be homogeneous, some transformations are required.

Example

We will solve the equation

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Multiplication of the numerator and denominator of the right-hand side of the equation by 1/x2 yields

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This above equation is homogeneous.

SECOND ORDER DIFFERENTIAL EQUATION

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In order to solve a nonhomogeneous of the second order linear differential equation we have to find a general solution of the corresponding homogeneous equation and then make use of either the method of variation of parameters or the method of undetermined coefficients to find the particular solution.

Example

We will solve the following initial value problem

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By the preceding example, the general solution is of the form

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We determine the partial solution by use of the method of undetermined coefficients. We guess y1 = Ax2 + Bx +C. Hence y1'' = 2A and

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This is satisfied for any x if and only if A = 1, B = 0, C = -2. Then y1 = x2 - 2, and the general solution is of the form

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Applying initial conditions we get the constants C1 and C2. We calculate the first derivative

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so

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Hence C2 = 2, C1 = 1 and the actual solution satisfying the initial conditions reads

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The expression "solve an equation" should be understand as a task of determining the general solution of this equation.



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