Question

Find the general solution of the following equations. $$y^{\prime}(x)=-y+2$$

Short answer

Answer: The general solution of the first-order linear differential equation $$y^{\prime}(x)=-y+2$$ is $$y(x)=2+\frac{C}{e^{x}}.$$

Step-by-step solution

Rewrite the differential equation in standard form

The given equation is $$y^{\prime}(x)=-y+2.$$ We can rewrite this equation in the standard form for first-order linear differential equations:$$y^{\prime}(x)+y=2.$$

Identify the integrating factor

We see that the coefficient of the $$y$$ term in our equation is $$1.$$ The integrating factor for this differential equation is $$\mu(x)=e^{\int 1 dx}=e^{x}.$$

Multiply both sides of the equation by the integrating factor

Multiplying both sides of the equation by $$e^{x}$$ gives:$$e^{x}y^{\prime}(x)+e^{x}y=2e^{x}.$$

Notice that the left side is a derivative

The left side of the equation is the derivative of the product of the integrating factor and dependent variable: $$\frac{d}{dx}(e^{x}y)=2e^{x}.$$

Integrate both sides of the equation

Now we integrate both sides of the equation with respect to $$x$$:$$\int \frac{d}{dx}(e^{x}y) dx=\int 2e^{x} dx.$$ This will give us the equation:$$e^{x}y(x)=2e^{x}+C.$$

Solve for y(x)

Finally, we isolate $$y(x)$$ by dividing both sides of the equation by $$e^{x}:$$ $$y(x)=\frac{2e^{x}+C}{e^{x}}.$$ Since $$\frac{2e^{x}}{e^{x}}=2$$ and $$\frac{C}{e^{x}}$$ is an arbitrary constant, we can write the general solution as: $$y(x)=2+\frac{C}{e^{x}}.$$

Key concepts

Integrating Factor

A first-order linear differential equation can be solved using the method of integrating factors. This technique simplifies the process of finding a solution. The idea behind an integrating factor is to multiply the entire equation by a special function that makes the left-hand side recognizable as a derivative. This allows integration to proceed simply.
To determine the integrating factor, take a look at the standard form of a linear differential equation: \[y'(x) + P(x)y = Q(x).\]* **Identify the coefficient**: The coefficient of the $$y$$ term, denoted as $$P(x)$$, is crucial. For our equation, $$P(x) = 1$$.* **Calculate the integrating factor**: It is given by: \[\mu(x) = e^{\int P(x) \, dx}.\] In our example, we find the integrating factor as: \[\mu(x) = e^{x}.\]Multiplying the entire differential equation by this integrating factor transforms it into a simpler equation. This step is critical, as it sets up the equation for straightforward integration.

General Solution

The general solution of a differential equation represents a family of solutions incorporating all possible particular solutions. When using the integrating factor method, once we've identified the right integrating factor, we can express the general solution in a streamlined way.
Here's how it works:1. **Multiply by the integrating factor**: After determining the integrating factor, multiply it across the equation. This gathers terms conveniently together.2. **Recognize the derivative**: Often, the left side becomes the derivative of a product, simplifying the equation substantially.3. **Integrate**: Integrate both sides to find a solution.In our specific problem, integrating both sides after recognizing the left-hand side as a derivative gives:\[\frac{d}{dx}(e^{x}y) = 2e^{x}.\]Integrating this equation results in:\[e^{x}y(x) = 2e^{x} + C.\]4. **Solve for $$y(x)$$**: Finally, solve this equation for the general solution of $$y(x)$$. This leads to:\[y(x) = 2 + \frac{C}{e^{x}}.\]This expression represents the general solution since it contains the arbitrary constant $$C$$, encompassing all potential particular solutions.

Separation of Variables

While separation of variables is an alternate method to solve differential equations, it's essential to recognize when or why it may not be applicable. Separation of variables involves rearranging a differential equation so that different types of variables appear on opposite sides.
However, this method only works under certain conditions:- **Applicability**: It applies well to equations where variables can be completely separated.- **Example Structure**: A separable equation looks like \[\frac{dy}{dx} = g(x)h(y).\] Here, we can move all terms involving $$y$$ to one side and $$x$$ to the other.In our given problem, separating terms of $$dy$$ and $$dx$$ completely isn't feasible directly due to the linearity of the equation and its coefficients. Consequently, using the integrating factor method was more straightforward here, as it made the relationship between the terms clearer without requiring separation. Understanding why other methods might not simplify our equation as effectively is crucial in mastering differential equation solutions.