You will need

- - notebook;
- - handle

Instruction

1

If the equation is written in the form: dy/dx = q(x)/n(y) belong to the category of differential equations with separable variables. They can be solved by writing the condition in the differentials according to the following scheme: n(y)dy = q(x)dx. Then integrate both parts. In some cases the decision is recorded in the form of integrals, taken from known functions. For example, if dy/dx = x/y, you get q(x) = x, n(y) = y. Write it in the form ydy = xdx and integrate. You should get y^2 = x^2 + c.

2

Linear

**equations**consider the equation "first degree." Unknown function with its derivatives included in such equation only in the first degree. Linear differential equation has the form dy/dx + f(x) = j(x), where f(x) and g(x) is a function depending on x. The solution is written with integrals, is taken from the well-known functions.3

Note that a differential equation is equation of second order (contains second derivatives) Thus, for example, is the equation of simple harmonic motion written in the form of the General formula: md 2x/dt 2 = –kx. Such equations have, in General, private decisions. The equation of simple harmonic motion is an example of the important class of linear differential equations with constant coefficient.

4

Let's consider a more common example (second order) equations, where y and z are predetermined constant, f(x) is a given function. Such equations can be solved in different ways, for example, by means of integral transformations. The same can be said about linear equations of higher order with constant coefficients.

5

Please note that the equations that contain the unknown functions and their derivatives standing in a degree above the first, called non-linear. The solution of nonlinear equations is quite complex and therefore, is used for each your particular case.

# Advice 2 : How to determine the form of the differential equation

To determine the differential equations required in order to choose the appropriate each case the method of solution. Classification of species is quite large, and the decision is based on the methods of integration.

Instruction

1

The necessity of differential equations arises when the well-known properties of the function, and she remains an unknown quantity. Often this situation occurs in the study of physical processes. Function properties are described by its derivative, or differential, so the only way the problem is integration. Before proceeding to the solution, we need to determine the form of the differential equation.

2

There are several types of differential equations, the simplest of them is the expression y’ = f(x), where y’ = dy/DX. In addition, this type can be given equality f(x)•y’ = g(x), i.e. y’ = g(x)/f(x). Of course, this is only possible provided that f(x) becomes zero. Example: 3^x•y’ = x2 – 1 → y’ = (x2 - 1)/3^H.

3

Differential equation with separated variables are so named because the derivative of y’ in this case literally split in two components dy and DX, which are on opposite sides of the equal sign. This equation of the form f(u)•dy = g(x)•DX. Example: (Y2 – sin y)•dy = tg x/(x - 1)•DX.

4

Two of the described type of differential equations are called ordinary or abbreviated as ODE. However, the first order equation can be more complex, heterogeneous. They are called LIDE – linear inhomogeneous equation u’ + f(x)•y = g(x).

To LIDE includes, in particular, the Bernoulli equation y’ + f(x)•y = g(x)•y^a. Example: 2•y’ – x2•y = (ln x/X3)•Y2. And the equation in total differentials f(x, y)DX + g(x, y)dy = 0, where ∂FX(x, y)/∂y = ∂do(x, y)/∂X. Example: (X3 – 2•x•y)DX – х2dу = 0, where X3 – 2•x•y – differentiation with respect to x of the function ¼ •x^4 – x2• + C a (x2) – its differentiation with respect to y.

To LIDE includes, in particular, the Bernoulli equation y’ + f(x)•y = g(x)•y^a. Example: 2•y’ – x2•y = (ln x/X3)•Y2. And the equation in total differentials f(x, y)DX + g(x, y)dy = 0, where ∂FX(x, y)/∂y = ∂do(x, y)/∂X. Example: (X3 – 2•x•y)DX – х2dу = 0, where X3 – 2•x•y – differentiation with respect to x of the function ¼ •x^4 – x2• + C a (x2) – its differentiation with respect to y.

5

The simplest form of an ODE of second order is y’ + p•a’ + q•u = 0, where p and q are constant coefficients. LIDE of the second order is a complicated version of the ODE, namely u’ + p•a’ + q•y = f(x). Example: y’ – 5•u’ + 13•y = sin x. once p and q are functions of the argument x, the equation might look like this: y’ – 5•x2•y’ + 13•(x - 1)•y = sin X.

6

Higher orders differential equations are divided into three subspecies: admitting a reduction of order, equations with constant coefficients and with coefficients as functions of the argument x:

• The expression f(x, y^(m)^(m+1),...,^(n)) = 0 do not contain derivatives of lower order m, then, by replacing z= y^(m) it is possible to reduce the order. Then the equation converts to the form f(x, z, z’,..., z^(n - m)) = 0. Example:’’•x – 4•Y2 = y’ - 2 → z’•x – 4•Y2 = z - 2 where z = y’ = dy/DX;

• LOU u^(k) + p_(k-1)•y^(k-1) + ... + p1•y’ + p0•u = 0 and LIDE have^(k) + p_(k-1)•y^(k-1) + ... + p1•y’ + p0•y = f(x) with constant coefficients pi. Examples: y^(3) + 2•’ – 15•y’ + 3•y = 0 and y^(3) + 2•’ – 15•y’ + 3•y = 2•X3 – ln x;

• LOU u^(k) + p(x)_(k-1)•y^(k-1) + ... + p1(x)•y’ + p0(x)•y = 0 and LIDE have^(k) + p(x)_(k-1)•y^(k-1) + ... + p1(x)•y’ + p0(x)•y = f(x) with coefficients-functions pi(x). Examples: y’’ + 2•x2•’ – 15•agsp x•y’ + 9•x•y = 0 and y’’ + 2•x2•’ – 15•arcsin x•y’ + 9•x•y = 2•X3 – ln Kh.

• The expression f(x, y^(m)^(m+1),...,^(n)) = 0 do not contain derivatives of lower order m, then, by replacing z= y^(m) it is possible to reduce the order. Then the equation converts to the form f(x, z, z’,..., z^(n - m)) = 0. Example:’’•x – 4•Y2 = y’ - 2 → z’•x – 4•Y2 = z - 2 where z = y’ = dy/DX;

• LOU u^(k) + p_(k-1)•y^(k-1) + ... + p1•y’ + p0•u = 0 and LIDE have^(k) + p_(k-1)•y^(k-1) + ... + p1•y’ + p0•y = f(x) with constant coefficients pi. Examples: y^(3) + 2•’ – 15•y’ + 3•y = 0 and y^(3) + 2•’ – 15•y’ + 3•y = 2•X3 – ln x;

• LOU u^(k) + p(x)_(k-1)•y^(k-1) + ... + p1(x)•y’ + p0(x)•y = 0 and LIDE have^(k) + p(x)_(k-1)•y^(k-1) + ... + p1(x)•y’ + p0(x)•y = f(x) with coefficients-functions pi(x). Examples: y’’ + 2•x2•’ – 15•agsp x•y’ + 9•x•y = 0 and y’’ + 2•x2•’ – 15•arcsin x•y’ + 9•x•y = 2•X3 – ln Kh.

7

The specific differential equation is not always obvious. Then you should carefully review it to bring to one of the canonical types to apply the appropriate method of solution. This can be done by different methods, the most common ones are replacement and decomposition into components of the derivative y’ = dy/DX.