{"id":4707,"date":"2014-02-10T06:05:45","date_gmt":"2014-02-09T21:05:45","guid":{"rendered":"http:\/\/fujiitoshiki.com\/improvesociety\/?p=4707"},"modified":"2014-08-01T18:58:09","modified_gmt":"2014-08-01T09:58:09","slug":"general-linear-differential-equation-of-order-n","status":"publish","type":"post","link":"https:\/\/www.fujiitoshiki.com\/improvesociety\/?p=4707","title":{"rendered":"General linear differential equation of order n<\/em>"},"content":{"rendered":"
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The general linear differential equation of order n<\/em> has the form<\/p>\n

\\displaystyle a_0(x)\\frac{d^ny}{dx^n} + a_1(x)\\frac{d^{n-1}y}{dx^{n-1}} + \\cdots + a_{n-1}(x)\\frac{dy}{dx} + a_n(x)y = R(x)\\ \\ \\ (1)<\/p>\n

A differential equation which cannot be written in this form is called nonlinear<\/em>. <\/p>\n

\\displaystyle x\\frac{d^2y}{dx^2} + 3\\frac{dy}{dx} - 2xy = \\sin x is a second order linear equation. \\displaystyle y\\frac{d^2y}{dx^2} - x\\left(\\frac{dy}{dx}\\right)^2 + x^2y = e^{-x} is a second order nonlinear equation. <\/p>\n

If R(x)<\/em>, the right side of (1), is replaced by zero the resulting equation is called the complementary, reduced<\/em> or homogeneous<\/em> equation. If R(x)<\/em> ≠ 0, the equation is called the complete<\/em> or nonhomogeneous<\/em> equation. <\/p>\n

If \\displaystyle x\\frac{d^2y}{dx^2} + 3\\frac{dy}{dx} -2xy =\\sin x is the complete equation, then \\displaystyle x\\frac{d^2y}{dx^2} + 3\\frac{dy}{dx} - 2xy = 0 is the corresponding complementary, reduced or homogeneous equation. <\/p>\n

If a_0(x)\\, \\cdots \\,a_n(x) are all constants, (1) is said to have constant coefficient<\/em>, otherwise it is said to have variable coefficients<\/em>. <\/p>\n<\/blockquote>\n