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First-Order Linear Equations
First-Order Linear Equations
A first-order linear equation has the standard form $$y'+P(x)y=Q(x).$$
Integrating Factor
Use $$\mu(x)=e^{\int P(x)dx}.$$ Multiply the equation by $\mu(x)$ to make the left side a product derivative.
Example 1
Solve $y'+y=0$.
Integrating factor is $e^x$. This gives $$\frac{d}{dx}(e^xy)=0,$$ so $e^xy=C$ and $$y=Ce^{-x}.$$
Example 2
Solve $y'+2y=4$.
Integrating factor: $e^{2x}$. Then $$\frac{d}{dx}(e^{2x}y)=4e^{2x}.$$ Integrating gives $$e^{2x}y=2e^{2x}+C,$$ so $$y=2+Ce^{-2x}.$$
Practice Problems
1. Identify $P(x)$ in $y'+3y=x$.
2. What is the integrating factor for $y'+2y=5$?
3. Solve $y'+y=0$.
4. Solve $y'+2y=0$.
5. Why use an integrating factor?
Show Answer Key
1. $3$
2. $e^{2x}$
3. $y=Ce^{-x}$
4. $y=Ce^{-2x}$
5. It turns the left side into the derivative of a product