It is necessary to define a type of the differential equation to pick up the way of the decision corresponding to each case. Classification of types quite big, and the decision is based on integration methods.
Instruction
1. Need for the differential equations arises when properties of function are known, and it remains unknown size. Often such situation arises at a research of physical processes. Properties of function are described by its derivatives or differential therefore the only way of its stay is integration. Before starting the decision, it is necessary to define a type of the differential equation.
2. There are several types of the differential equations, the simplest of them is expression at’ = f (x) where at’ = dy/dx. Besides, equality of f (x) can be brought to this look • at’ = g (x), i.e. at’ = g (x)/f (x). Certainly, it is possible only provided that f (x) does not address in zero. Example: 3^х • at’ = x² – 1 → at’ = (x² - 1)/3^х.
3. The differential equations with the divided variables are called so because the derivative at’ in this case is literally divided into two components of du and dx which are on different sides from the sign equally. It equations of a type of f (y) • du = g (x) • dx. Example: (at² – sin y) • du = tg x / (x - 1) • dx.
4. Two described types of the differential equations carry the name ordinary or for short the ODE. However the equations of the first order can be and more difficult, non-uniform. They are called LNDU – the linear non-uniform equations at’ + f (x) • at = g (x). Bernoulli's equation at’ + f (x) belongs to LNDU, in particular • at = g (kh) ·y^a. Example: 2 · at’ – x ²\• at = (ln x/x³) • at². And also the equation in full differentials of f (x, s) dx + g (x, s) du = 0, where fx (x, s) / u = gu (x, s) / kh. Example: (x³ – 2 • x • s) dx – x²dу = 0, where x³ – 2 • x • at – a private derivative on x from function ¼\• х^4 – x ²\• at + C, and (– x²) – its private derivative on at.
5. The simplest look the ODE of the second order is at’’ + p • at’ + q • at = 0, where p and q – constant coefficients. LNDU of the second order is the complicated version the ODE, namely at’’ + p • at’ + q • at = f (x). Example: at’’ – 5 • at’ + 13 • at = sin x. If p and q – functions of an argument x, then the equation can look approximately so: at’’ – 5 • x ²\• at’ + 13 • (x - 1) • at = sin x.
6. The differential equations of the highest orders are subdivided into three subspecies: allowing decrease in an order, equation with constant coefficients and with coefficients in the form of functions of an argument x: • Expression of f (x, у^ (m), у^(m+1), …, у^ (n)) = 0 does not contain derivatives below m order, so through replacement of z = у^ (m) can reduce an order. Then the equation will be transformed to a type of f (x, z, z’, …, z^ (n - m)) = 0. Example: at’’’ • x – 4 • at² = at’ - 2 → z’’ • x – 4 • at² = z - 2, where z = at’ = dy/dx; • LODU у^ (k) + p _ (k-1)·y^ (k-1) + … + p1 • at’ + p0 • at = 0 and LNDU у^ (k) + p _ (k-1)·y^ (k-1) + … + p1 • at’ + p0 • at = f (x) with constant coefficients of pi. Examples: у^ (3) + 2 • at’’ – 15 • at’ + 3 • at = 0 and у^ (3) + 2 • at’’ – 15 • at’ + 3 • at = 2 • x³ – ln x; • LODU у^ (k) + p (x) _ (k-1)·y^ (k-1) + … + p1 (x) • at’ + p0 (x) • at = 0 and LNDU у^ (k) + p (x) _ (k-1)·y^ (k-1) + … + p1 (x) • at’ + p0 (x) • at = f (x) with coefficients functions of pi (x). Examples: at ''' + 2 • x ²\• at’’ – 15·arssin x • at’ + 9 • x • at = 0 and at’’’ + 2 • x ²\• at’’ – 15•arcsin x • at’ + 9 • x • at = 2 • x³ – ln x.
7. The type of the concrete differential equation not always is obvious. Then it is necessary to consider attentively it regarding reduction to one of initial types to apply the corresponding way of the decision. It is possible to make it different methods, the most widespread of them are replacement and decomposition of a derivative by components at’ = dy/dx.