In Problems 1-22 Solve The Given Differential Equation By Separation Of Variables. 1. Dy/dx = Sin 5x
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Introduction to Differential Equations and Separation of Variables
Differential equations are fundamental in mathematics and applied sciences, describing how quantities change with respect to each other. They are equations involving derivatives, which represent rates of change. Among various methods to solve differential equations, separation of variables is a powerful technique, especially suitable for equations where variables can be algebraically separated on different sides of the equation.
In this article, we will focus on solving the differential equation:
\[ \frac{dy}{dx} = \sin 5x \]
using the separation of variables method. This particular equation is a straightforward example that illustrates the core principles of the method and demonstrates how to approach similar problems efficiently.
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Understanding the Differential Equation: \(\frac{dy}{dx} = \sin 5x\)
Before diving into the solution process, let's analyze the structure of the differential equation:
- The derivative \(\frac{dy}{dx}\) is expressed explicitly in terms of the independent variable \(x\).
- The right-hand side, \(\sin 5x\), is a function solely of \(x\).
Since the equation is separable, we can think of it as:
\[ dy = \sin 5x \, dx \]
which indicates that the change in \(y\) depends only on \(x\), and the equation can be integrated directly once separated.
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Step-by-Step Solution Using Separation of Variables
Step 1: Rewrite the Differential Equation
Begin by expressing the differential equation in a form that isolates \(dy\) and \(dx\):
\[ \frac{dy}{dx} = \sin 5x \]
which can be rewritten as:
\[ dy = \sin 5x \, dx \]
This form indicates that the differential change in \(y\) is directly proportional to \(\sin 5x\).
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Step 2: Integrate Both Sides
To find the general solution, integrate both sides:
\[ \int dy = \int \sin 5x \, dx \]
The left side integrates straightforwardly:
\[ y = \int \sin 5x \, dx + C \]
where \(C\) is the constant of integration.
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Step 3: Compute the Integral \(\int \sin 5x \, dx\)
The integral of \(\sin 5x\) with respect to \(x\) can be computed using substitution:
- Let \(u = 5x\), then \(du = 5 dx \Rightarrow dx = \frac{du}{5}\).
Rewriting the integral:
\[ \int \sin 5x \, dx = \int \sin u \cdot \frac{du}{5} = \frac{1}{5} \int \sin u \, du \]
The integral:
\[ \int \sin u \, du = -\cos u + D \]
Putting it all together:
\[ \int \sin 5x \, dx = -\frac{1}{5} \cos u + D \]
Substituting back \(u = 5x\):
\[ \int \sin 5x \, dx = -\frac{1}{5} \cos 5x + D \]
Since \(D\) is an arbitrary constant, it can be absorbed into the overall constant \(C\).
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Step 4: Write the General Solution
Combining the results:
\[ y = -\frac{1}{5} \cos 5x + C \]
This is the general solution to the differential equation, representing a family of functions parameterized by the constant \(C\).
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Interpreting the Solution
The solution:
\[ y(x) = -\frac{1}{5} \cos 5x + C \]
has several notable features:
- It involves a cosine function scaled by \(\frac{1}{5}\), indicating the amplitude of oscillation.
- The variable \(C\) shifts the entire graph vertically, representing initial conditions or specific solutions.
- The period of the cosine function is:
\[ T = \frac{2\pi}{5} \]
which is derived from the coefficient of \(x\) inside the cosine, indicating how frequently the oscillations occur.
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Applications of the Solution
Understanding and solving such differential equations has wide applications in fields like physics, engineering, biology, and economics.
Some practical scenarios include:
- Oscillatory systems: Modeling systems with periodic behavior, such as pendulums or electrical circuits.
- Population dynamics: When growth or decline depends on periodic factors.
- Signal processing: Analyzing waveforms and oscillations.
The explicit solution allows for precise analysis of these systems, especially when initial conditions are known.
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Additional Examples and Practice Problems
To solidify understanding, here are similar problems that employ separation of variables:
- Solve \(\frac{dy}{dx} = \cos 3x\)
- Solve \(\frac{dy}{dx} = e^{2x}\)
- Solve \(\frac{dy}{dx} = y \sin x\)
- Solve \(\frac{dy}{dx} = \frac{1}{x}\)
Each problem involves integrating the right-hand side after separation, emphasizing the versatility of the method.
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Summary and Key Takeaways
- Separation of variables is a straightforward method for solving differential equations where variables can be separated on different sides.
- The key steps include rewriting the differential equation, integrating both sides, and applying initial conditions if available.
- The solution to \(\frac{dy}{dx} = \sin 5x\) is \( y = -\frac{1}{5} \cos 5x + C \).
- Recognizing the structure of the differential equation simplifies the solution process.
- These solutions are crucial in modeling periodic phenomena and systems exhibiting oscillatory behavior.
Conclusion
Mastering the separation of variables technique enables students and professionals to solve a broad class of differential equations efficiently. The problem \(\frac{dy}{dx} = \sin 5x\) exemplifies how direct integration transforms a differential equation into a functional relationship, providing insights into the behavior of the modeled system. Practice with similar problems will deepen understanding and enhance problem-solving skills in differential equations.
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For further learning, consider exploring differential equations with more complex functions, initial conditions, and applications in various scientific fields.
