“homogeneous first‐order differential equation”的概念、定义、翻译、参考文献-科学参考

homogeneous first‐order differential equation

Mathematics

A first‐order differential equation dy/dx = f(x,y) in which the function f, of two variables, has the property that f(kx,ky) = f(x,y) for all k. Examples of such functions are

$\frac{x^{2} + 3 y^{2}}{2 x^{2} - 5 x y}, 1 + e^{x / y}, \frac{x}{\sqrt{x^{2} + y^{2}}} .$

Any such function f can be written as a function of one variable v, where v = y/x. The method of solving homogeneous first‐order differential equations is to let y = vx so that dy/dx = x(dv/dx) + v. The differential equation for v as a function of x that is obtained is always separable.

单词	homogeneous first‐order differential equation
释义	homogeneous first‐order differential equation Mathematics A first‐order differential equation dy/dx = f(x,y) in which the function f, of two variables, has the property that f(kx,ky) = f(x,y) for all k. Examples of such functions are $\frac{x^{2} + 3 y^{2}}{2 x^{2} - 5 x y}, 1 + e^{x / y}, \frac{x}{\sqrt{x^{2} + y^{2}}} .$ Any such function f can be written as a function of one variable v, where v = y/x. The method of solving homogeneous first‐order differential equations is to let y = vx so that dy/dx = x(dv/dx) + v. The differential equation for v as a function of x that is obtained is always separable.
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