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Magazine Chapter 2: Problem 29
$$ \text { In Problems } 29 \text { and } 30 \text {, solve the given initial- value problem. } $$ $$ \frac{d y}{d x}=\cos (x+y), \quad y(0)=\pi / 4 $$
Short Answer
Expert verified
Use substitution \( z = x+y \), separate variables to get \( \frac{1}{2}\tan(z/2) = x + C \), and solve using initial condition \( y(0) = \frac{\pi}{4} \).
Step by step solution
01
Understand the Initial Value Problem
We are given a differential equation \( \frac{dy}{dx} = \cos(x + y) \) with the initial condition \( y(0) = \frac{\pi}{4} \). Our goal is to find the function \( y(x) \) satisfying both the differential equation and the initial condition.
02
Analyze the Differential Equation
The differential equation \( \frac{dy}{dx} = \cos(x + y) \) is non-linear and separable. Separating the variables involves rearranging it to look like \( y' = f(x+y) \). This suggests a substitution technique since it doesn't directly separate.
03
Substitute Variables
Consider substituting \( z = x + y \), then the derivative \( \frac{dz}{dx} = 1 + \frac{dy}{dx} \). Substituting gives \( \frac{dz}{dx} = 1 + \cos(z) \).
04
Solve the Separable Equation
Rewrite the equation as \( \frac{1}{1 + \cos(z)} dz = dx \). This equation can now be integrated with respect to \( z \) and \( x \).
05
Integrate Both Sides
Integrate \( \int \frac{1}{1 + \cos(z)} dz = \int dx \). This requires using trigonometric identities to simplify, such as \( 1 + \cos(z) = 2\cos^2(z/2) \), leading to \( \int \frac{1}{2\cos^2(z/2)} dz = \int dx \).
06
Simplify Using Trigonometric Identities
Recognize that \( \frac{1}{\cos^2(z/2)} = \sec^2(z/2) \), which integrates to \( \tan(z/2) \). Thus, we have \( \frac{1}{2} \tan(z/2) = x + C \).
07
Apply Initial Condition
Using the initial condition \( y(0) = \frac{\pi}{4} \), find \( z = 0 + \frac{\pi}{4} = \frac{\pi}{4} \) at \( x = 0 \). Substitute into the equation: \( \frac{1}{2}\tan(\frac{\pi}{8}) = 0 + C \). Solve for \( C \).
08
Solve for the Particular Solution
Plug back the constant \( C \) into the equation and express \( \tan(z/2) \) back in terms of \( y \), using \( z = x + y \), to find \( y(x) \).
09
Rearrange and Simplify
Rearrange to find \( y = x + [\mathrm{expression involving } x \text{ and constants]} \). Simplify further to obtain \( y(x) \) explicitly.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Differential Equations
In mathematics, a differential equation is an equation that involves an unknown function and its derivatives. These equations play a critical role in modeling various real-world phenomena where change is involved.
Understanding the basics:
- Differential equations can be classified according to their order, linearity, and homogeneity.
- The order of a differential equation is determined by the highest derivative present.
- An initial value problem specifies not only the differential equation but also an initial condition, helping to pinpoint a specific solution from a family of possible solutions.
Our original problem involves a first-order differential equation combined with an initial condition. This requires you to not only solve the equation but also meet a given starting point for the function.
Separable Equations
A separable equation is a kind of differential equation where the variables can be separated from each other. In other words, you can rearrange the equation such that each side contains only one variable.
Key points for understanding separable equations:
- Begin by organizing terms involving one variable (for example, all terms including 'x') on one side and terms involving the other variable (here, 'y') on the other side.
- Once separated, each side of the equation can be integrated independently.
Our example equation, though initially not directly separable, was manipulated using substitution techniques to achieve a separable form. This highlights the interconnected nature of these mathematical strategies.
Trigonometric Identities
Trigonometric identities are equations that relate different trigonometric functions to one another. They are crucial tools in solving complex calculus problems, especially those involving derivatives and integrals of trigonometric functions.Important things to remember about trigonometric identities:- Common identities include reciprocal identities, Pythagorean identities, and angle sum/difference identities.- These identities often simplify equations, making integration or differentiation much more manageable.In this problem, using the identity \(1 + \cos(z) = 2\cos^2(z/2)\) transformed a challenging integral into a more approachable format. This allowed the integration process to proceed smoothly.
Substitution Techniques
Substitution is a powerful method for solving differential equations. It involves replacing one variable with another, simplifying the equation into a more manageable form. This method is particularly useful when direct separation of variables isn't possible.Steps to consider when applying substitution:- Identify a substitution that simplifies the equation, such as letting \(z = x + y\) in our current problem.- By expressing derivatives in terms of the new variable, the equation often simplifies and becomes easier to integrate.- Finally, after solving the substituted equation, substitute back to get the original variables.In the exercise, substituting \(z = x + y\) simplified our non-linear differential equation, converting it into a form where standard integration techniques were applicable.
