Verifying Solutions for Differential Equations · 验证微分方程的解
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| solution/səˈluːʃn/ | 解 | jiě |
Is this function actually a solution?
- Before solving a differential equation, you should be able to check a proposed answer.
- A solution 解 of a differential equation is a function that makes the equation true.
- The test is purely mechanical: differentiate the candidate and substitute.
- If both sides match for all $x$, it's a solution; if not, it isn't.
这个函数真的是解吗?
- 在解微分方程之前,你应该能检验一个提议的答案。
- 微分方程的解是使方程成立的函数。
- 这个检验纯粹是机械的:对候选求导并代入。
- 若两边对所有 $x$ 都相符,它就是解;否则不是。
The verification recipe
- 1. Take the candidate function and compute its derivative(s).
- 2. Substitute the function and its derivatives into the differential equation.
- 3. Simplify both sides and check that they are equal (as identities in $x$).
- No integration needed — verifying is just differentiating and plugging in.
验证配方
- 1. 取候选函数,算出它的导数。
- 2. 把函数及其导数代入微分方程。
- 3. 化简两边,检查它们是否相等(作为关于 $x$ 的恒等式)。
- 无需积分——验证只是求导并代入。
To verify a candidate solution, you... · 要验证候选解，你需要...
Differentiate the candidate, substitute, check both sides match. · 对候选解求导，代入，检查两边是否相等。
A worked check
- Does $y=e^{3x}$ solve $\dfrac{dy}{dx}=3y$?
- Differentiate: $\dfrac{dy}{dx}=3e^{3x}$.
- Substitute into the right side: $3y=3e^{3x}$.
- Left $=$ right ($3e^{3x}=3e^{3x}$), so yes, $y=e^{3x}$ is a solution. ✓
一个验证范例
- $y=e^{3x}$ 是否满足 $\dfrac{dy}{dx}=3y$?
- 求导:$\dfrac{dy}{dx}=3e^{3x}$。
- 代入右边:$3y=3e^{3x}$。
- 左 $=$ 右($3e^{3x}=3e^{3x}$),所以是,$y=e^{3x}$ 是解。✓
Solutions of dy/dx = ky · dy/dx = ky 的解
Each exponential $y=Ce^{kx}$ threads the slope field — verifying one just confirms its derivative matches $ky$. · 每条指数曲线$y=Ce^{kx}$穿过方向场——验证其中一个即确认其导数匹配$ky$。
$y=e^{3x}$ is a solution of $\dfrac{dy}{dx}=3y$. · $y=e^{3x}$是$\dfrac{dy}{dx}=3y$的解。
$\frac{dy}{dx}=3e^{3x}=3y$. ✓
Both sides must be equal for ____ values of $x$, not just one, to confirm a solution. · 对于____值的$x$，两边必须相等，而不仅仅是一个，才能确认解。
A solution satisfies the equation identically. · 解恒等满足方程。
General vs. particular solutions
- $y=Ce^{3x}$ (with any constant $C$) also solves $\dfrac{dy}{dx}=3y$ — the whole family is the general solution.
- Pick a specific $C$ and you get a particular solution.
- Verifying works the same for either: differentiate and check the equation holds.
- The constant carries through the differentiation, so the check still balances.
通解与特解
- $y=Ce^{3x}$(带任意常数 $C$)也满足 $\dfrac{dy}{dx}=3y$——整个族是通解。
- 选一个具体的 $C$,就得到一个特解。
- 验证对两者都一样:求导并检查方程成立。
- 常数在求导中被带过去,所以检验仍然平衡。
Verifying a given solution requires integrating the differential equation. · 验证给定解需要对微分方程积分。
Verifying is differentiate-and-substitute; no integration. · 验证即求导并代入；无需积分。
For which constants $C$ does · 作用 $y=Ce^{3x}$ solve $\dfrac{dy}{dx}=3y$? · 对于哪些常数$C$，$y=Ce^{3x}$是$\dfrac{dy}{dx}=3y$的解？
$\frac{dy}{dx}=3Ce^{3x}=3y$ for all $C$ — the general solution. · $\frac{dy}{dx}=3Ce^{3x}=3y$对应所有$C$——通解。
Is $y=x^2+C$ a solution of $\dfrac{dy}{dx}=2x$? · $y=x^2+C$是否是$\dfrac{dy}{dx}=2x$的解？
$\frac{dy}{dx}=2x$ regardless of $C$. · $\frac{dy}{dx}=2x$无论$C$为何值。
Verifying is not solving — you're only checking a given candidate, so no integration is involved. Substitute the function and its derivative into the equation and confirm both sides are identical for all $x$, not just at one point. A match at a single $x$ isn't enough.
验证不是求解——你只是检查一个给定的候选,所以不涉及积分。把函数及其导数代入方程,确认两边对所有 $x$ 都相同,而非仅在一个点。在单个 $x$ 相符是不够的。
Verify that $y=x^2+C$ solves $\dfrac{dy}{dx}=2x$.
- Differentiate: $\dfrac{dy}{dx}=2x$ (the $+C$ vanishes).
- The differential equation says $\dfrac{dy}{dx}=2x$.
- Both sides equal $2x$, so $y=x^2+C$ is a solution for every constant $C$. ✓
验证 $y=x^2+C$ 满足 $\dfrac{dy}{dx}=2x$。
- 求导:$\dfrac{dy}{dx}=2x$($+C$ 消失)。
- 微分方程说 $\dfrac{dy}{dx}=2x$。
- 两边都等于 $2x$,所以对每个常数 $C$,$y=x^2+C$ 都是解。✓
To verify a solution of a differential equation, differentiate the candidate and substitute it (with its derivatives) into the equation; it's a solution iff both sides are equal for all $x$. No integration required — verifying is differentiate-and-check. A family with a constant $C$ is the general solution.
要验证微分方程的解,对候选求导并把它(连同导数)代入方程;它是解当且仅当两边对所有 $x$ 相等。无需积分——验证就是求导并检查。带常数 $C$ 的族是通解。