Problem 16 - Calculus Single Variable

Question

Solve the given differential equation. $$ \cot (x) \frac{d y}{d x}=\tan (y) $$

Step-by-Step Solution

Answer

The solution is \( \sin(y) = K \sin(x) \) with constant \( K \).

Step 1: Rewrite the Equation

Start by rearranging the given differential equation: \( \cot(x) \frac{dy}{dx} = \tan(y) \). To simplify, rewrite it as \( \frac{dy}{dx} = \tan(y) \cot(x) \).

Step 2: Separate Variables

To separate the variables, rewrite the equation as \( \frac{dy}{\tan(y)} = \cot(x) \, dx \). This separates the \( y \) and \( x \) terms on either side of the equation.

Step 3: Integrate Both Sides

Integrate both sides of the equation separately. The left side becomes \( \int \frac{dy}{\tan(y)} = \int \cot(y) \, dy = \ln |\sin(y)| + C_1 \) and the right side becomes \( \int \cot(x) \, dx = \ln |\sin(x)| + C_2 \).

Step 4: Set the Integrals Equal

Equate the two integral expressions: \( \ln |\sin(y)| + C_1 = \ln |\sin(x)| + C_2 \). Subtract \( C_1 \) from both sides to get \( \ln |\sin(y)| = \ln |\sin(x)| + C \) where \( C = C_2 - C_1 \).

Step 5: Exponentiate to Remove Logarithms

Remove the logarithms by exponentiating both sides of the equation: \( |\sin(y)| = A |\sin(x)| \), where \( A = e^C \), thus representing a new constant.

Step 6: Simplify the Expression

Express the final solution in terms of \( y \) and \( x \) with the constant: \( \sin(y) = K \sin(x) \), where \( K \) represents a new constant equal to \( \pm A \).

Key Concepts

Separation of Variables

Differential equations can often seem complex, but separating variables is a handy technique to simplify them. In our given exercise, we began with a differential equation involving functions of both \( x \) and \( y \). The goal of separation of variables is to rearrange the equation so that all terms involving \( y \) are on one side and all terms involving \( x \) are on the other.
Essentially, we moved from \( \cot(x) \frac{dy}{dx} = \tan(y) \) to \( \frac{dy}{\tan(y)} = \cot(x) \, dx \). This crucial step allows us to handle each variable separately by integrating both sides independently."
This method works particularly well when the equation can be easily written in terms of distinct functions of \( x \) and \( y \). It's important to ensure each side is dimensionally consistent, meaning you have differentials \( dy \) and \( dx \) aligning appropriately post-separation. This transforms the task into evaluating simpler, one-variable integrals.

Integration Techniques

Once we've separated the variables, the next step involves integrating both sides of the equation. Here, integration techniques come into play. For our exercise, we needed to integrate \( \frac{dy}{\tan(y)} \) and \( \cot(x) \, dx \). These integrations utilize the following formulas:

• \( \int \cot(y) \, dy = \ln |\sin(y)| + C_1 \)
• \( \int \cot(x) \, dx = \ln |\sin(x)| + C_2 \)

These integrals are solved using standard trigonometric integrals. Remember, integration typically introduces a constant of integration—denoted here as \( C_1 \) and \( C_2 \).
By understanding the basic integral forms and applying them correctly, solving the separate integrals becomes a smoother process. The constants of integration are adjusted later to fit specific boundary or initial conditions of the problem.

Trigonometric Identities

Trigonometric identities are powerful tools for simplifying and solving equations like ours. When working with trigonometric functions, such identities can transform an equation into a more workable form. In our case, after separating and integrating, we dealt with expressions like \( \ln |\sin(y)| = \ln |\sin(x)| + C \).
Exponentiating both sides helped eliminate logarithms, leading to the expression \( |\sin(y)| = A |\sin(x)| \), with \( A = e^C \) as a constant. Subsequently, acknowledging that sine function ranges encompass positive and negative values, allows finding a relationship like \( \sin(y) = K \sin(x) \).
Utilizing these identities and understanding the properties of trigonometric functions helps simplify expressions and solve equations more efficiently. Knowledge of identities such as the Pythagorean identity or double-angle formulas can often be useful when managing more complex trigonometric differential equations.