Integrating Factors for First- and Higher-Order Equations

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What is an integrating factor?

A function multiplied into a differential equation to make its terms combine into a derivative that can be integrated.

For $y''=Ay^{2/3}$, what first integral follows after multiplying by $y'$ and integrating?

$(y')^2=\frac{6A}{5}y^{5/3}+C_0$.

What integrating factor solves a first-order linear equation $y'+P(x)y=Q(x)$?

$M(x)=e^{\int P(x)\,dx}$, which makes the left side $(My)'$.

When does the second-order integrating-factor pattern apply?

When the equation has the form $y''+2py'+(p^2+p')y=h$, so multiplying by $M=e^{\int p\,dx}$ gives $(My)''=Mh$.

Why is the product-derivative integrating-factor method restrictive at higher orders?

The coefficients must match the specific combination produced by differentiating $M(x)y$ repeatedly.

Quiz questions

1. For the first-order linear equation $y'+P(x)y=Q(x)$, which integrating factor makes the left side the derivative of a product?Answer: $e^{\int P(x)\,dx}$
2. If $M(x)$ is an integrating factor for a first-order linear equation, why does multiplying $M$ by a nonzero constant still give an integrating factor?Answer: The constant multiplies both sides and cancels when solving for the dependent variable.
3. For the equation $y'-\frac{2}{x}y=0$, why are solutions considered on intervals that do not contain $x=0$?Answer: The coefficient $-2/x$ is undefined at $x=0$.
4. Which coefficient pattern allows a second-order equation to become $(My)''=Mh$ after multiplication, with $M=e^{\int p(x)\,dx}$?Answer: $y''+2p(x)y'+(p(x)^2+p'(x))y=h(x)$
5. For an $n$th-order equation that matches the derivative pattern of $(My)^{(n)}$, what is the next step after multiplication by $M$?Answer: Integrate $n$ times and then divide by $M$.