Chapter 8: Problem 11

Find the "general solution" (that is, a solution containing an arbitrary constant) of each of the following differential equations, by separation of variables. Then find a particular solution of each equation satisfying the given boundary conditions. \(\cos x \cos y d x-\sin x \sin y d y=0, \quad y=\pi\) when \(x=\pi / 2\)

Short Answer

Expert verified

The general solution is defined by |\text{sin}(y)| = K |\text{cos}(x)|, using the condition \text{it does need to be rechecked}

Step by step solution

Rewrite the Differential Equation

Start by separating variables. Rewrite the given differential equation ewline ewline \[ \frac{\text{cos}(x) \text{cos}(y) \text{d}x - \text{sin}(x) \text{sin}(y) \text{d}y = 0}{\text{cos}(x) \text{cos}(y) \text{d}x}= \text{sin}(x) \text{sin}(y) \text{d}y \] ewline ewline to ewline ewline \[ \frac{\text{cos}(y) \text{d}x}{\text{sin}(y)} = \frac{\text{sin}(x) \text{d}y}{\text{cos}(x)} \] ewline ewline

Separate the Variables

Bring all terms involving x to one side of the equation and terms involving y to the other side: ewline ewline \[ \text{cot}(y) \text{d}y = \frac{\text{sin}(x)}{\text{cos}(x)} \text{d}x \] ewline ewline Expressing this using integrals gives ewline ewline \[ \text{cot}(y) \text{d}y = \text{tan}(x) \text{d}x \]

Integrate Both Sides

Integrate both sides to separate the variables: ewline ewline \[ \text{ln}|\text{sin}(y)| = - \text{ln}|\text{cos}(x)| + C i where C is the constant of integration \] ewline ewline Using the property of logarithms, we can rewrite this as ewline ewline \[ |\text{sin}(y)| = K |\text{cos}(x)| \text{where} i K = \text{e}^{C} \]

Solve for y in Terms of x

Rewriting the absolute values back and solving for y(x) ewline ewline \[ \text{sin}(y) = K \text{cos}(x) \] ewline ewline This is the general solution in terms of x and y with an arbitrary constant K.

Find the Particular Solution

Using the given boundary condition \ y = \text{pi} \ when \ x = \frac{\text{pi}}{2} \ substitute these into the general solution to find K. ewline ewline \[ \text{sin}(\text{pi}) = K \text{cos}\frac{\text{pi}}{2} \] ewline since \ \text{sin(pi) = 0} \text{and} \text{cos} \frac{\text{pi}}{2} = 0 \ this implies \ K = \text{undefined} \

Summary of the solutions

Therefore the simplified versions

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Differential Equations

A differential equation involves a function and its derivatives. They appear in many disciplines such as physics, engineering, and economics.

A common type is the **ordinary differential equation (ODE)**, where the function has one independent variable. Our exercise gives an ODE formed by \(\text{cos}(x) \text{cos}(y) \text{d}x - \text{sin}(x) \text{sin}(y) \text{d}y = 0\).

Differential equations often describe how variables change over time or space. The goal is to find a solution that satisfies both the equation and any given conditions.

**General Solution**: A solution containing arbitrary constants.
**Particular Solution**: A solution from the general form that meets specific boundary conditions.

Boundary Conditions

Boundary conditions restrict solutions to make them unique. They give information at specific points of the domain.

In our exercise, the boundary condition is \(\begin{cases} y = \text{pi} \text{ when} x = \frac{\text{pi}}{2} \).

To find a particular solution:

Insert the boundary values into the general solution
Solve for any constants involved

Here, using the boundary condition, we substitute \(\text{sin}(\text{pi}) = K \text{cos}(\frac{\text{pi}}{2})\) into our general solution \( \text{sin}(y) = K \text{cos}(x)\).

This helps determine \( K\) for a specific solution, making the general form comply with the problem's criteria.

Integration

Integration is a fundamental calculus operation. It connects differential equations to their solutions by finding functions whose derivatives match given expressions.

In step 3 of our solution, we integrate both sides to separate variables:

Left side: \( \text{cot}(y) \text{d}y \)
Right side: \( \text{tan}(x) \text{d}x \)

These integrals lead to:
\[ \text{ln}|\text{sin}(y)| = -\text{ln}|\text{cos}(x)| + C \] where \( C \) is the constant of integration.

**Logarithms** simplify the relation, allowing us to find a relationship between \( y \) and \( x \) in terms of \( K = e^C \).
Finally, rewriting and solving the absolute values helps us form a tractable general solution.