The differential equation into which unknown function and its derivative are included linearly that is in the first degree, is called the linear differential equation of the first order.
Instruction
1. General view of the linear differential equation of the first order is as follows: y ′ + p (x) *y = f(x) where y is unknown function, and p(x) and f(x) — some set functions. They are considered as continuous in that area in which it is required to integrate the equation. In particular, they can be also constants.
2. If f(x) ≡ 0, then call the equation uniform; if is not present — that, respectively, non-uniform.
3. The linear uniform equation can be solved by method of division of variables. Its general view: y ′ + p (x) *y = 0, therefore: dy/dx = - p (x) *y from where follows that dy/y = - p (x) dx.
4. Integrating both parts of the turned-out equality, we receive: ∫(dy/y) = - ∫p(x)dx, that is ln(y) = - ∫p(x)dx + ln(C) or y = C*e^ (-∫p(x)dx)).
5. The solution of the non-uniform linear equation can be output from the uniform relevant decision, that is the same equation with the rejected right Part f (x). For this purpose it is necessary to replace constant C in the solution of the uniform equation with unknown function φ (x). Then the solution of the non-uniform equation will be presented in the form: y = φ (x) *e^ (-∫ p (x) dx)).
6. Differentiating this expression, we will receive that derivative of y it is equal: y ′ = φ ′ (x) *e^ (-∫p(x)dx) - φ (x) *p(x) * e^ (-∫p(x)dx). Having substituted the found expressions for y and y ′ in the initial equation and having simplified received, it is easy to come to result: dφ/dx = f(x) *e^ (∫p(x)dx).
7. After integration of both parts of equality it receives a look: φ (x) = ∫ (f(x) *e^ (∫ p (x) dx)) dx + C1. Thus, required function y will be expressed in a look: y = e^ (-∫p(x)dx) * (C + ∫f(x) *e^ (∫ p (x) dx)) dx).
8. If to equate a constant C to zero, then from expression for y it is possible to receive the private solution of the set equation: y1 = (e^ (-∫p(x)dx)) * (∫f(x) *e^ (∫ p (x) dx)) dx). Then the full decision can be expressed in a look: y = y1 + C*e^ (-∫p(x)dx)).
9. In other words, the full solution of the linear non-uniform differential equation of the first order is equal to the sum of its private decision and the common decision of the corresponding uniform linear equation of the first order.