How to find the integrating factor?
Understand the Problem
The question is asking how to determine the integrating factor used in solving first-order linear differential equations. The process typically involves identifying the standard form of the equation and applying the formula for the integrating factor based on the coefficient of the dependent variable.
Answer
The integrating factor is $I(x) = e^{\int P(x) \, dx}$.
Answer for screen readers
The integrating factor is given by the formula $$ I(x) = e^{\int P(x) , dx} $$ where $P(x)$ is the coefficient of $y$ in the standard form of the linear differential equation.
Steps to Solve
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Identify Standard Form First, we need to make sure the differential equation is in the standard form, which is $$ \frac{dy}{dx} + P(x)y = Q(x) $$ where $P(x)$ and $Q(x)$ are functions of $x$.
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Extract P(x) From the standard form mentioned above, identify the function $P(x)$ that is the coefficient of $y$.
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Calculate the Integrating Factor The integrating factor, denoted as $I(x)$, is calculated using the formula: $$ I(x) = e^{\int P(x) , dx} $$ Perform the integration of $P(x)$ with respect to $x$, then exponentiate the result.
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Example of the Integrating Factor For example, if $P(x) = 2x$, then we find $$ I(x) = e^{\int 2x , dx} = e^{x^2} $$
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Now you have the Integrating Factor You can now use this integrating factor to solve the original differential equation.
The integrating factor is given by the formula $$ I(x) = e^{\int P(x) , dx} $$ where $P(x)$ is the coefficient of $y$ in the standard form of the linear differential equation.
More Information
The integrating factor is a crucial tool that simplifies the process of solving first-order linear differential equations, allowing us to make the equation exact and easier to manipulate.
Tips
- Forgetting to put the equation in standard form before identifying $P(x)$.
- Miscomputing the integral of $P(x)$ when finding the integrating factor. Always double-check your integration.
