Problem 3 - Calculus Early Transcendentals

Question

Solve the differential equation by the method of integrating factors. $$ y^{\prime}+y=\cos \left(e^{x}\right) $$

Step-by-Step Solution

Answer

The solution to the differential equation is \( y = e^{-x} \sin(e^x) + Ce^{-x} \).

Step 1: Write the differential equation in standard form

The given equation is \( y^{\prime} + y = \cos(e^x) \). It is already in the standard linear form of \( y' + P(x)y = Q(x) \) with \( P(x) = 1 \) and \( Q(x) = \cos(e^x) \).

Step 2: Determine the integrating factor

The integrating factor \( \mu(x) \) is obtained using the formula: \[ \mu(x) = e^{\int P(x)\, dx} = e^{\int 1\, dx} = e^x. \]

Step 3: Multiply the entire equation by the integrating factor

Multiply both sides of the differential equation by \( \mu(x) = e^x \):\[ e^x y' + e^x y = e^x \cos(e^x). \]

Step 4: Recognize the left-hand side as a derivative

The left side of the equation \( e^x y' + e^x y \) is the derivative of the product \( \frac{d}{dx}(e^x y) \). Rewrite the equation as:\[ \frac{d}{dx}(e^x y) = e^x \cos(e^x). \]

Step 5: Integrate both sides

Integrate both sides with respect to \( x \):\[ \int \frac{d}{dx}(e^x y) \, dx = \int e^x \cos(e^x) \, dx. \]This simplifies to:\[ e^x y = \int e^x \cos(e^x) \, dx + C, \]where \( C \) is the constant of integration.

Step 6: Solve the integral on the right side

The integral \( \int e^x \cos(e^x) \, dx \) is solved by substitution. Let \( u = e^x \), then \( du = e^x dx \). The integral becomes:\[ \int \cos(u) \, du = \sin(u) + C_1 = \sin(e^x) + C_1. \]

Step 7: Write the general solution

Substitute back to the equation from step 5:\[ e^x y = \sin(e^x) + C. \]Solve for \( y \):\[ y = e^{-x}(\sin(e^x) + C). \]

Step 8: Finalize the solution

The general solution of the differential equation is:\[ y = e^{-x} \sin(e^x) + Ce^{-x}, \]where \( C \) is an arbitrary constant.

Key Concepts

Integrating Factors

Integrating factors come into play when solving linear first-order differential equations. These equations often appear in the form \( y' + P(x)y = Q(x) \). The integrating factor is a crucial mathematical tool that helps in transforming a differential equation into a more easily solvable form.

To calculate the integrating factor, we use the formula:

• \( \mu(x) = e^{\int P(x) \, dx} \)

This factor simplifies the process by enabling us to convert the left side of the differential equation into a single derivative. Once we have the integrating factor, we multiply it throughout the equation. This step helps in easily recognizing and integrating the expression.

In our example, the integrating factor turns out to be \( e^x \) since \( P(x) = 1 \). Recognizing the integrating factor is vital, as it leads us toward finding the solution by integrating over both sides. The beauty of this method lies in its ability to unravel complex expressions into more manageable forms.

Linear Differential Equations

Linear differential equations form a significant class of equations, characterized by the linearity of the unknown function and its derivatives. They neatly fit the form \( y' + P(x)y = Q(x) \), where the function \( P(x) \) and \( Q(x) \) do not involve \( y \) or its derivatives. These equations are prevalent in various fields like physics and engineering due to their structured nature.

Linear differential equations are approachable because they allow specific solution methods, such as integrating factors or variation of parameters. With the solution path often well-defined, they offer fewer complexities compared to nonlinear equations.

In our example, \( y' + y = \cos(e^x) \), the equation is clearly linear with \( P(x) = 1 \) and \( Q(x) = \cos(e^x) \). Recognizing a problem as a linear differential equation allows us to apply systematic approaches, making them invaluable to mathematicians and scientists alike.

Solution Methods

Solution methods for differential equations give us strategies to tackle these sometimes tricky problems effectively. For linear differential equations, one of the most reliable methods is the use of integrating factors.

The step-by-step solution often involves:

• Recognizing the equation type and writing it in standard linear form.
• Identifying the correct integrating factor.
• Rewriting the equation and solving the resulting expressions by integration.

Integrating both sides of the equation is key to finding the general solution. In cases where a substitution can simplify an integral, it can be a helpful additional method.

In our example, after integrating and substitution, we derive at the solution:

• \[ y = e^{-x} \sin(e^x) + Ce^{-x} \]

This general solution incorporates the constant \( C \), reflecting unknowns dependent on initial conditions. Solution methods are all about methodically untangling the problem to identify a clear path to the answer.