# Integrate 1st Order Diff EQ: Sin2x - 2Cos2x + c

In summary, the conversation is about using a formula for first order differential equations and integrating to find the solution. The solution is y(x) = e^(x) (int[e^(-x)*-sin(2x)dx] + c), which simplifies to y(x)= 1/5(sin2x-2cos2x)+c. The participants also discuss the differentiation of the answer and using integration by parts.
howdy, i uhh solved this problem early in the semester, and now I am looking at it like wtf did i do to get the right answer...it says use the 'off the shelf formula' for first order diff eqns.

its pretty much just integrating now

y(x) = e^(x) (int[e^(-x)*-sin(2x)dx] + c)

ans => y(x)= 1/5(sin2x-2cos2x)+c

Are you sure you transcribed those correctly? What do you get when you differentiate the answer you've listed? Just sin and cos terms, right?

If $$y(x) = e^{x}\int e^{-x}\times -\sin 2x \; dx$$

ya that's what i don't understand..

use integration by parts.

Ignore the ex outside the integral until you have done the integral itself, using integration by parts (twice) as courtigrad said. (Presumably the integration will have a factor of e-x, canceling the ex outside the integral.)

## What is the purpose of integrating a first order differential equation?

Integrating a first order differential equation allows us to find the original function that satisfies the given differential equation. It helps us solve problems involving rates of change and can be applied to various fields such as physics, engineering, and economics.

## What does the notation "Sin2x - 2Cos2x + c" represent in this differential equation?

The notation "Sin2x - 2Cos2x + c" represents the general solution to the given first order differential equation. The "c" is a constant that is added to account for all possible solutions to the equation.

## How do you solve for "c" in the general solution?

To solve for "c", we need to use initial conditions. These are specific values given in the problem that allow us to determine the exact solution. We substitute these values into the general solution and solve for "c".

## Can this differential equation be solved using separation of variables?

Yes, this differential equation can be solved using separation of variables. This method involves separating the variables on each side of the equation and then integrating both sides to find the general solution.

## Are there any real-world applications for this type of differential equation?

Yes, this type of differential equation can be applied to various real-world problems. For example, it can be used to model the growth of a population, the decay of a radioactive substance, or the movement of a falling object due to gravitational force.

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