Recipe

Solve a first-order linear equation

Use when: the equation is (or can be rearranged into) \dfrac{dy}{dx}+P(x)y=Q(x) — y and y' appear only to the first power, never multiplied together or raised to a power

Goal: an explicit solution y(x)

Taught in First-order linear equations and integrating factors

1. 1

   Divide through so the coefficient on y' is exactly 1, giving \dfrac{dy}{dx}+P(x)y=Q(x).

   Example

   Take \dfrac{dy}{dx}+\dfrac2xy=x^2 (for x>0) — already standard form, since the coefficient on y' is already 1. Reading off P(x)=\dfrac2x and Q(x)=x^2 is only safe because that coefficient is 1; a leading 3\dfrac{dy}{dx} would need to be divided away first.

   Why: P(x) only means what the method needs it to mean once the y' coefficient is 1 — skipping this step is the most common way the method goes wrong.

2. 2

   Compute \mu(x)=e^{\int P(x)\,dx}.

   Example

   \mu(x)=e^{\int (2/x)\,dx}=e^{2\ln x}=x^2.

   Why: \mu(x) is chosen for exactly one purpose — multiplying the standard-form equation by it collapses the entire left side into the single derivative \frac{d}{dx}[\mu(x)y].

3. 3

   Multiply both sides by \mu(x), then rewrite the left side as \dfrac{d}{dx}[\mu(x)y].

   Example

   x^2\dfrac{dy}{dx}+2xy=x^4, and the left side is exactly \dfrac{d}{dx}[x^2y], so \dfrac{d}{dx}[x^2y]=x^4

   Why: this is the identity the integrating factor exists to create — it turns an equation that mixes y and y' into one that's directly integrable.

4. 4

   Integrate to get \mu(x)y=\displaystyle\int\mu(x)Q(x)\,dx.

   Example

   x^2y=\int x^4\,dx=\dfrac{x^5}{5}+C

   Why: the left side is now literally the derivative of \mu(x)y, so undoing the derivative is the only step left on that side.

5. 5

   Divide by \mu(x) to isolate y; if given y(x_0), substitute it to solve for the constant.

   Example

   Dividing by x^2: y=\dfrac{x^3}{5}+\dfrac{C}{x^2}. If y(1)=3: 3=\dfrac15+C, so C=\dfrac{14}5, giving \boxed{y=\dfrac{x^3}{5}+\dfrac{14}{5x^2}}

   Why: dividing by \mu(x) (never zero, since it's an exponential) is what finally isolates the actual unknown.