Question

Find the general solution of each of the following differential equations. \(y^{\prime}+y=e^{x}\)

Short answer

The general solution is \( y = \frac{1}{2} e^x + Ce^{-x} \).

Step-by-step solution

Identify the form of the differential equation

The differential equation given is a first-order linear differential equation of the form \(y' + P(x)y = Q(x)\). Here, \(P(x) = 1\) and \(Q(x) = e^{x}\).

Find the integrating factor

The integrating factor \( \mu(x) \) is calculated using the formula \( \mu(x) = e^{\int P(x) dx} \).Plugging in \(P(x) = 1\), we get \( \mu(x) = e^{\int 1 dx} = e^x \).

Multiply the differential equation by the integrating factor

Multiply both sides of the differential equation by the integrating factor \(e^x\):\( e^x y' + e^x y = e^x e^x \). This simplifies to \( e^x y' + e^x y = e^{2x} \).

Simplify the left-hand side

Notice that the left-hand side \( e^x y' + e^x y \) can be written as the derivative of \( y \cdot e^x\):\( \frac{d}{dx}(y e^x) = e^{2x} \).

Integrate both sides

Integrate both sides with respect to x:\( \int \frac{d}{dx}(y e^x)dx = \int e^{2x} dx \).This simplifies to \( y e^x = \int e^{2x} dx \).

Perform the integration

Integrate \( \int e^{2x} dx \):\( \int e^{2x} dx = \frac{1}{2} e^{2x} + C \), where C is the constant of integration.So, \( y e^x = \frac{1}{2} e^{2x} + C \).

Solve for y

Divide both sides by \( e^x\) to solve for y:\( y = \frac{1}{2} e^x + Ce^{-x} \).

Key concepts

Integrating Factor

First-order linear differential equations often require special techniques to solve. One of these methods is using an integrating factor. An integrating factor is a function that simplifies the equation, making it easier to solve.

For a differential equation of the form:
\( y' + P(x)y = Q(x) \),
the integrating factor, denoted as \(\bf{\textmu(x)} \), is found using the formula:
\[ \textmu(x) = e^{\int P(x) \, dx} \]
Here, we see that our given differential equation is \( y' + y = e^x \). Identifying \(P(x) = 1 \) and \(Q(x) = e^x \), we calculate the integrating factor:
\[ \textmu(x) = e^{\int 1 \, dx} = e^{x} \]
This function \(e^x\) will make our differential equation easier to work with.

General Solution

After calculating the integrating factor, the next step in solving the differential equation is to use it to find the general solution.

We do this by multiplying the entire differential equation by the integrating factor \(e^x\):

\( e^x y' + e^x y = e^{x} e^x \)
Simplifies to: \( e^x y' + e^x y = e^{2x} \)
Left-hand side can be written as the derivative: \( \frac{d}{dx}(y e^x) = e^{2x} \)

Now, we integrate both sides with respect to x:
\[ \int \frac{d}{dx}(y e^x)dx = \int e^{2x} dx \]
Which simplifies to: \( y e^x = \int e^{2x} dx \)

Performing the integration, we get:
\[ \int e^{2x} dx = \frac{1}{2}e^{2x} + C \]
Thus, the solution in its intermediate form is: \( y e^x = \frac{1}{2} e^{2x} + C \)

Finally, we solve for y:
\( y = \frac{1}{2} e^x + Ce^{-x} \)
This represents the general solution of the differential equation.

Differential Equation Integration

Integration is a fundamental technique to solve differential equations. After transforming the equation using the integrating factor, we often encounter integrals we need to evaluate:

Consider the equation after applying the integrating factor \(e^x\):
\( \frac{d}{dx}(y e^x) = e^{2x} \)
This tells us that the left-hand side is the derivative of \( y e^x \), so to eliminate the derivative, we integrate both sides:

\[ \int \frac{d}{dx}(y e^x) dx = \int e^{2x} dx \]

The integral on the left-hand side becomes \( y e^x \), while the integral on the right side is performed as follows:
\[ \int e^{2x} dx = \frac{1}{2}e^{2x} + C \]

Thus resulting in:
\( y e^x = \frac{1}{2} e^{2x} + C \)
Finally, to solve for y, we divide both sides by \(e^x\), simplifying to:
\(y = \frac{1}{2} e^x + Ce^{-x} \)
This demonstrates how integrating both sides facilitates finding the solution to the differential equation. Integration helps turn the transformed equation into a clearer, explicit solution for y.