Problem 26 - Advanced Engineering Mathematics

Question

Each \(D E\) in Problems 23-30 is of the form given in (5). In Problems 23-28, solve the given differential equation by using an appropriate substitution. $$ \frac{d y}{d x}=\sin (x+y) $$

Step-by-Step Solution

Answer

The general solution is given by \( x + y = 2 \tan^{-1}(e^{x+C}) \).

Step 1: Identify the Substitution

The differential equation is \( \frac{dy}{dx} = \sin(x+y) \). Notice that \( x+y \) appears inside the sine function, suggesting we use the substitution \( v = x + y \). This substitution will help simplify the expression.

Step 2: Differentiate the Substitution

Using the substitution \( v = x + y \), we differentiate both sides with respect to \( x \). Thus, we have \( \frac{dv}{dx} = 1 + \frac{dy}{dx} \).

Step 3: Rewrite the Original DE

We know \( \frac{dy}{dx} = \sin(x+y) \) can be rewritten as \( \frac{dy}{dx} = \sin(v) \) under our substitution. Substitute into the differentiated equation to get \( \frac{dv}{dx} = 1 + \sin(v) \).

Step 4: Solve the New Differential Equation

Separate variables in the equation \( \frac{dv}{dx} = 1 + \sin(v) \) to get \( \frac{dv}{1 + \sin(v)} = dx \). Integrate both sides with respect to their respective variables.

Step 5: Integrate and Solve

The left-hand side requires a trigonometric identity for \( \sin(v) \). Recognize that \( 1 + \sin(v) = (\sin(v/2) + \cos(v/2))^2 \), so perform the substitution \( t = \tan(v/2) \), and use trig identities to integrate. For the right-hand side, integrate to yield \( x + C \). Solve for \( v \) in terms of \( x \) and \( C \).

Step 6: Substitute Back the Original Variables

Re-substitute \( v = x + y \) back in terms of \( x \) and solve for \( y \), the original dependent variable. This yields the general solution to the given differential equation.

Key Concepts

Substitution Method

The substitution method is a powerful tool used to simplify differential equations by introducing a new variable. This technique often helps reduce the complexity of an equation by transforming it into a more familiar form. In the exercise given, the differential equation \( \frac{dy}{dx} = \sin(x+y) \) suggests a substitution because the expression \( x+y \) appears together inside the sine function.

By substituting \( v = x + y \), we effectively transform the problem into a single-variable equation. This substitution allows us to reframe the differential equation with respect to \( v \) instead of the original variables. Consequently, when we differentiate \( v = x + y \), the equation becomes \( \frac{dv}{dx} = 1 + \frac{dy}{dx} \).

This step is crucial because it aligns the given differential equation with our substitution, turning it into \( \frac{dv}{dx} = 1 + \sin(v) \). The substitution method can often simplify an equation significantly, enabling more straightforward solutions.

Separation of Variables

Once a differential equation is simplified using a substitution, the separation of variables can then be applied. This technique is used when an equation can be restructured to have all instances of one variable on one side, and the other variable on the opposite side.

In the example provided, after substituting and transforming, we have \( \frac{dv}{dx} = 1 + \sin(v) \). To apply separation of variables, we rearrange this into \( \frac{dv}{1+\sin(v)} = dx \).

Separation of variables relies on the ability to isolate each variable facilitating easier integration. By integrating both sides separately, we can find a relation between the variables, unraveling the solution step-by-step. This method works best for problems where variables can be neatly divided. Integrating leads us forward in solving for \( v \) which we later back-substitute to find the original solution.

Trigonometric Identities

Trigonometric identities are mathematical expressions that allow trigonometric functions to be rewritten in other forms. They are essential in solving differential equations involving these functions. In our exercise, the integral on the left was \( \int \frac{dv}{1+\sin(v)} \).

To solve this, we recognize an opportunity to utilize a trigonometric identity. The identity \( 1 + \sin(v) = \left(\cos(v/2) + \sin(v/2)\right)^2 \) aids in simplifying and integrating the expression.

By substituting \( t = \tan(v/2) \), the expression can be rewritten using this identity. Trigonometric identities like these are invaluable for converting challenging trigonometric integrals into simpler forms, making the process of integration achievable. Ultimately, understanding and applying the appropriate identities allows us to solve the trigonometric part of the differential equation effectively.