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

FIRST ORDER DIFFERENTIAL EQUATION

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.

Figure 1

Example:

The general solution of

is:

See the next Figure:

FIRST ORDER DIFFERENTIAL EQUATION WITH SEPARABLE VARIABLES

The graph of the solution:

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

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

General solution is

The required particular solution is

UNDETERMINED COEFFICIENTS

The method of undetermined coefficients of solving a nonhomogeneous equation

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:

• a polynomial,

• a sum α sinω x + β cosω x,

• a function ae^bx, and b ≠ -p,

• a sum or a product of the types listed above.

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

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

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

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

Since

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

hence

Substituting back into the equation we get

that is

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

Hence

Thus the function

is the desired particular solution, which implies that

is the general solution of the equation

.

Example

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

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) = xe^x .

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

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 e^x

Hence

Thus

This condition is satisfied if and only if

that is, for any x

Hence

which gives

Thus the particular solution is of the form

whereas the general solution is

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

The desired particular solution is

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

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

Indeed, differentiating the equality

we get

Substituting in the equation

Separation of variables yields

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

Substituting

we obtain

and further

For f (u) ≠ u, that is, if
we separate variables and integrate

That is

For f (u) = u we have

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

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

This above equation is homogeneous.

SECOND ORDER DIFFERENTIAL EQUATION

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

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

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

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

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

so

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

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

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