Solving differential equations : Differential equation, such as dy / dx = x / y, does not comply with a number and function, in this particular case is such that its graph at any point, for example, at coordinates (2,3) has a tangent with slope equal to relative coordinates (in our example 2 / 3). This is easily seeing, if we construct a large number of points and from each delay a short period with a corresponding slope. How to solve differential equations? -The solution would be a function whose graph regards each point of the corresponding segment. If the points and segments enough, we can roughly outline the shape of the curves-solutions (three of these curves are shown in Fig. 1). There is exactly one curve of the solution passing through each point with y № 0. Each individual solution is called a particular solution of differential equations, if we can find a formula that contains all the partial solutions (except, perhaps, a few individuals), then we say that the general solution. Particular solution is a single function, whereas in common - their whole family. How to solve differential equations - Solve a differential equation - it means to find either its private or a general solution. In our example, the general solution is y² – x²= c, where c - any number, a particular solution passing through the point (1,1) has the form y = x and is obtained at c = 0; particular solution passing through the point (2,1) has the form y² – x²= 3. Requiring that the curve is a solution took place, for example, through the point (2, 1), called the initial condition (as sets the starting point on the curve solving).

y² – x² =4; y² – x²=0; y² – x²=-4

We can show that in the example (1) general solution is x = ce^–kt, where c - constant, which can be determined, for example, specifying the amount of the substance at t = 0. The equation of Example (2) - a special case of the example (1), corresponding to k = 1 / 100. The initial condition is x = 10 at t = 0 gives the particular solution x = 10e^–t/100. The equation of Example (4) has the general solution T = 70 + ce^–kt and a particular solution 70 + 130^–kt; to determine the value of k, more data needed.

Differential equation dy/dx = x / y is called the first-order equation; because it contains the first derivative (order differential equation is considered to be the order of entering into it the highest derivative). Most (though not all) emerging practice of differential equations of the first type pass through each point only one curve is a solution.

There are several important types of first order differential equations, which admit solutions in the form of formulas containing only elementary functions - power, exponents, logarithms, sinuses and cosines, etc. These equations are the following.

 - Equation with multiple variables.
- Exact differential equations.
- Linear Equations
- The equations of higher orders
- Nonlinear differential equations
- Existence theorems
- Partial differential equations