Finding General Solutions Using Separation of Variables · 用分离变量法求通解
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| separable/ˈsepərəbl/ | 可分离 | kě fēn lí |
Get the $x$'s and $y$'s on opposite sides
- To actually solve a differential equation (not just sketch it), we integrate — but first we sort the variables.
- A differential equation is separable 可分离 if you can write it so one side has only $y$ and the other only $x$.
- $\dfrac{dy}{dx}=g(x)\,h(y)$ is the tell-tale form: a product of an $x$-part and a $y$-part.
- Then each side can be integrated on its own.
把 $x$ 和 $y$ 分到两边
- 要真正解一个微分方程(而非仅画草图),我们靠积分——但先把变量分开。
- 若能写成一边只含 $y$、另一边只含 $x$,微分方程就是可分离的。
- $\dfrac{dy}{dx}=g(x)\,h(y)$ 是标志形式:一个 $x$ 部分与一个 $y$ 部分的乘积。
- 然后每一边都能各自积分。
Which equation is separable? · 哪个方程是可分离的？
$xy$ factors as ($x$-part)$\times$($y$-part); a sum does not. · $xy$可分解为($x$-部分)$\times$($y$-部分)；和式则不能。
$\dfrac{dy}{dx}=x+y$ is not separable because you cannot split a ____. · $\dfrac{dy}{dx}=x+y$ 不可分离，因为你无法拆分一个 ____。
Separation needs a product, not a sum. · 分离变量需要乘积形式，而非求和形式。
Separate the variables
- Treat $\dfrac{dy}{dx}$ as a ratio of differentials and move each variable to its own side.
- From $\dfrac{dy}{dx}=g(x)\,h(y)$: divide by $h(y)$ and multiply by $dx$:
-
$$\frac{1}{h(y)}\,dy = g(x)\,dx$$
- Now each side involves only one variable and its differential — ready to integrate.
分离变量
- 把 $\dfrac{dy}{dx}$ 当作微分之比,把每个变量移到自己那一边。
- 由 $\dfrac{dy}{dx}=g(x)\,h(y)$:除以 $h(y)$、乘以 $dx$:
-
$$\frac{1}{h(y)}\,dy = g(x)\,dx$$
- 现在每一边只含一个变量及其微分——可以积分了。
Separating $\dfrac{dy}{dx}=xy$ gives... · 分离$\dfrac{dy}{dx}=xy$得到...
Divide by $y$, multiply by $dx$: $\tfrac1y\,dy=x\,dx$. · 除以$y$，乘以$dx$：$\tfrac1y\,dy=x\,dx$。
Integrate both sides
- Integrate the left in $y$ and the right in $x$: $\displaystyle\int\frac{1}{h(y)}\,dy=\int g(x)\,dx$.
- Add a single constant $+C$ (one is enough — combine both sides' constants).
- Then solve for $y$ if you can, to get the general solution.
- The result is a family of functions with the constant $C$.
对两边积分
- 左边对 $y$、右边对 $x$ 积分:$\displaystyle\int\frac{1}{h(y)}\,dy=\int g(x)\,dx$。
- 加一个常数 $+C$(一个就够——把两边的常数合并)。
- 若能,再解出 $y$,得到通解。
- 结果是带常数 $C$ 的一族函数。
The general solution is a family · 通解是一个族
Solving $\tfrac{dy}{dx}=ay$ gives $y=Ce^{ax}$ — one curve per constant, all fitting the same field. · 求解$\tfrac{dy}{dx}=ay$得到$y=Ce^{ax}$——每个常数对应一条曲线，均符合同一方向场。
Integrating $\dfrac{1}{y}\,dy = x\,dx$ gives... · 积分$\dfrac{1}{y}\,dy = x\,dx$得到...
$\int\tfrac1y\,dy=\ln|y|$; $\int x\,dx=\tfrac{x^2}{2}$; add $+C$. · $\int\tfrac1y\,dy=\ln|y|$；$\int x\,dx=\tfrac{x^2}{2}$；加上$+C$。
Keep the $+C$ (and solve for $y$)
- The constant of integration is what makes this a family — never drop it.
- Often you'll exponentiate or rearrange to isolate $y$; the $C$ transforms but stays.
- Leaving the answer as an implicit relation is fine if $y$ can't be cleanly isolated.
- A later initial condition (lesson 7.7) will nail down $C$.
保留 $+C$(并解出 $y$)
- 积分常数正是让它成为一族的原因——绝不丢弃。
- 你常会取指数或重排来孤立 $y$;$C$ 会变形但仍在。
- 若 $y$ 无法干净地孤立,把答案留作隐式关系也可以。
- 之后的初始条件(7.7 课)会确定 $C$。
You should include the constant $+C$ when solving a differential equation by separation. · 在用分离变量法解微分方程时，你应该包含常数$+C$。
The $+C$ gives the general (family of) solutions. · $+C$ 给出通解（解族）。
The general solution of $\dfrac{dy}{dx}=xy$ is... · $\dfrac{dy}{dx}=xy$ 的通解是...
Exponentiate $\ln|y|=\tfrac{x^2}{2}+C$ to get $y=A e^{x^2/2}$. · 对 $\ln|y|=\tfrac{x^2}{2}+C$ 取指数运算得到 $y=A e^{x^2/2}$。
Separation works only if the equation factors as (function of $x$) $\times$ (function of $y$). Something like $\frac{dy}{dx}=x+y$ is not separable — you can't split a sum. And don't forget the $+C$ after integrating: without it you have one curve, not the general family.
分离只在方程能因式分解为(关于 $x$ 的函数)$\times$(关于 $y$ 的函数)时有效。像 $\frac{dy}{dx}=x+y$ 这样的不可分离——你无法拆开一个和。而且积分后别忘了 $+C$:没有它你只有一条曲线,而非通解族。
Solve $\dfrac{dy}{dx}=xy$ (general solution).
- Separate: $\dfrac{1}{y}\,dy = x\,dx$.
- Integrate: $\ln|y| = \dfrac{x^2}{2}+C$.
- Exponentiate: $y = e^{x^2/2+C}=A\,e^{x^2/2}$ (writing $A=e^C$). General solution: $y=A\,e^{x^2/2}$.
解 $\dfrac{dy}{dx}=xy$(通解)。
- 分离:$\dfrac{1}{y}\,dy = x\,dx$。
- 积分:$\ln|y| = \dfrac{x^2}{2}+C$。
- 取指数:$y = e^{x^2/2+C}=A\,e^{x^2/2}$(记 $A=e^C$)。通解:$y=A\,e^{x^2/2}$。
Separation of variables solves a separable equation $\frac{dy}{dx}=g(x)h(y)$: rearrange to $\frac{1}{h(y)}\,dy=g(x)\,dx$, integrate both sides (one $+C$), and solve for $y$ to get the general solution. It only works when the equation factors into an $x$-part times a $y$-part.
分离变量法解可分离方程 $\frac{dy}{dx}=g(x)h(y)$:重排为 $\frac{1}{h(y)}\,dy=g(x)\,dx$,对两边积分(一个 $+C$),再解出 $y$ 得到通解。它只在方程能分解为 $x$ 部分乘 $y$ 部分时有效。