This section explains how to solve differential equations called first-order homogeneous forms.
Since the variables appearing in the equation have equal orders, the basic policy is to transform them into new variables in the form of fractions, and to attribute them to the separable equation.
In doing so, care should be taken to separate the cases in which the variable in the denominator is zero.
Definition
A differential equation that can be written as a function of \(frac{y}{x}\) or \(\frac{x}{y}\), as in
$$\frac{dy}{dx}=f\left(\frac{y}{x}\right)\tag{1}$$
when \(y\) is a function of \(x\), is called a first-order homogeneous form.
The term "first-order" means that the differential equation consists of only first‐order differentiation \(\frac{dy}{dx}\), while "homogeneous form" means that the order of \(x\) and \(y\) in each term is equal.
How to solve
Variable transformations are attributed to the separable equation.
Let \(\frac{y}{x}=z\) and hence \(y=xz\), you obtain
$$\frac{dy}{dx}=\frac{d}{dx}(xz)=z+x\frac{dz}{dx}$$
Substituting these into equation \((1)\),
$$z+x\frac{dz}{dx}=f(z)$$
$$\frac{dz}{dx}=\frac{1}{x}(f(z)-z)=P(x)Q(z)$$
which can be attributed to the separable equation.
The solution to the separable equation can be found at
Example
$$(7x+3y)\frac{dy}{dx}=-9x-5y$$
Let \(x\neq 0\) and divide both sides by \(x\) to convert to a first-order homogeneous form.
$$(7+3\frac{y}{x})\frac{dy}{dx}=-9-5\frac{y}{x}$$
\(z=\frac{y}{x}\) とおくと \(frac{dy}{dx}=z+xfrac{dz}{dx}) であり、これらを代入して
$$(7+3z)left(z+xfrac{dz}{dx}right)=-9-5z$$
$$(7+3z)xfrac{dz}{dx}=-3z^2-12z-9$$
$$frac{dz}{dx}=-frac{3}{x}frac{(z+3)(z+1)}{7+3z}$$
とすると、変数分離形に帰着できる。
したがって、 (C_1) を任意定数として
$$intfrac{7+3z}{(z+3)(z+1)}dz=-intfrac{3}{x}dx+C_1$$
$$intleft(frac{2}{z+1}+frac{1}{z+3}right)dz=-intfrac{3}{x}dx+C_1$$
$$2ln|z+1|+ln|z+3|=-3ln|x|+C_1$$
$$ln|(z+1)^2(z+3)x^3|=C_1$$
$$|(z+1)^2(z+3)x^3|=exp(C_1)$$
$$(z+1)^2(z+3)x^3=pmexp(C_1)$$
ここで、 (C=pmexp(C_1)) とおくと
$$(z+1)^2(z+3)x^3=C$$
(z=frac{y}{x}) を再代入して
$$(frac{y}{x}+1)^2(frac{y}{x}+3)x^3=C$$
$$(y+x)^2(y+3x)=C$$
この関数は (x=0) でも定義される。
微分方程式の解法一覧
その他の微分方程式の解法は、以下の記事を参照のこと。
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