TASK
- Provided a worked example for {{math concept}}
Requirements
- Respond with a wikitext code snippet (SimpleMathJax extension installed, use $$...$$ and $...$ delimiters)
IMPORTANT: think of reader
- primary wikireader: non-math-major undergraduates
- academic register
- concise language
- minimise technical jargons
- focus on core concept and basic applications
- do not try to include everything
Show, don't tell
- demonstrate with math, not words
- NO blabla about what readers can see for themselves in the math demonstrations.
- NO empty metacomments
- important steps shown in $$...$$ standalone math snippets
Style
- text book style writing
- no
=== title blocks or step 1,2,3 mechanical listing
Example
To see the method in action, consider
$$ \frac{dy}{dx} + \frac{1}{x}\,y = x^2, \qquad x>0. $$
Here $P(x)=1/x$, so
$$ \mu(x)=e^{\int \frac{1}{x}\,dx}=e^{\ln x}=x. $$
Multiply the whole equation by $\mu(x)=x$:
$$ x\frac{dy}{dx} + y = x^3. $$
The left side is exactly the derivative of the product $x\,y$:
$$ \frac{d}{dx}\bigl(xy\bigr) = x^3. $$
Integrate with respect to $x$:
$$ xy = \int x^3\,dx = \frac{x^4}{4} + C. $$
Finally, divide by $x$ to obtain the general solution:
$$ y = \frac{x^3}{4} + \frac{C}{x}. $$
A quick substitution verifies that this function family satisfies the original equation. The arbitrary constant $C$ can be determined later if an initial condition $y(x_0)=y_0$ is given.
