Problem 5 - Advanced Engineering Mathematics

Question

From \(\frac{1}{y} d y=\frac{4}{x} d x\) we obtain \(\ln |y|=4 \ln |x|+c\) or \(y=c_{1} x^{4}\).

Step-by-Step Solution

Answer

The general solution is \( y = C_1 x^4 \), where \( C_1 \) is a constant.

Step 1: Rearrange the Given Differential Equation

Starting with the equation \( \frac{1}{y} \, dy = \frac{4}{x} \, dx \), notice that each side can be integrated separately as both \( y \) and \( x \) are being treated independently in terms of their own variables.

Step 2: Integrate Both Sides

Integrate the left-hand side with respect to \( y \) and the right-hand side with respect to \( x \). The integrals become:\[ \int \frac{1}{y} \, dy = \int \frac{4}{x} \, dx \] The solution to these integrals are:\[ \ln |y| = 4 \ln |x| + C \] Where \( C \) is the constant of integration.

Step 3: Apply Logarithmic Properties

Using the properties of logarithms, the equation \( \ln |y| = 4 \ln |x| + C \) can be rewritten. By rewriting \( 4 \ln |x| \) as \( \ln |x^{4}| \), the equation becomes:\[ \ln |y| = \ln |x^{4}| + C \].Express \( C \) as \( \ln |C_1| \), where \( C_1 \) is a positive constant, leading to:\[ \ln |y| = \ln (|C_1 x^4|) \]

Step 4: Remove the Logarithms

Exponentiate both sides of the equation from Step 3 to eliminate the logarithms. This gives you:\[ |y| = |C_1 x^4| \]Since \( C_1 \) is a constant, set \( C_1 \) as a positive constant without the absolute value to simplify the expression. Therefore:\( y = C_1 x^4 \)

Step 5: Provide General Solution

The solution to the differential equation is now neatly expressed as:\[ y = C_1 x^4 \]where \( C_1 \) is a constant that results from the integration constant. This constant could represent a family of functions depending on initial conditions, but typically it's just a positive constant.

Key Concepts

Separation of Variables

The technique of separation of variables is a fundamental method used to solve a particular type of differential equation. It allows us to solve equations in the form of \( \frac{dy}{dx} = g(x)h(y) \). By isolating each variable on different sides of the equation, the variables are separated. In the given exercise, the equation \( \frac{1}{y} dy = \frac{4}{x} dx \) is appropriately aligned for using the separation of variables.

• We move all terms involving \( y \) to one side and terms involving \( x \) to the other.
• This gives us two smaller integrals that are easier to solve individually.

Understanding separation of variables makes solving first-order differential equations more manageable, leading directly to solutions by integration.

Integration

Integration is the next crucial step once the variables are separated, as it helps in finding functions for the original differential equation. In this exercise, after separating the variables, we integrate both sides individually:

• \( \int \frac{1}{y} \, dy \) gives the result \( \ln |y| \).
• \( \int \frac{4}{x} \, dx \) simplifies to \( 4 \ln |x| \).

These integrals solve independently, allowing us to construct a solution in terms of logarithms. Integration provides the means to transition from the differential to an equation, establishing a relationship between \( y \) and \( x \) for the problem. This step is a bridge from a dynamic situation to a static relationship by utilizing known integration techniques.

Logarithmic Properties

The integration phase often leads to logarithmic equations, as seen in this exercise. Here is where the properties of logarithms become very useful. After integrating, we get \[ \ln |y| = 4 \ln |x| + C \] Applying logarithmic rules can simplify this expression.

• Using the property \( a \ln b = \ln b^a \), change \( 4 \ln |x| \) to \( \ln |x^{4}| \).
• Rewriting \( C \) as a logarithm (\( \ln |C_1| \)) allows the equation to be expressed as \( \ln |y| = \ln (|C_1 x^4|) \).

Understanding these properties eases the process of solving equations involving logs, making exponentiation straightforward to remove the logs.

Constant of Integration

The constant of integration, often denoted as \( C \), is essential when solving indefinite integrals. It represents an infinite number of potential solutions that differ by a constant. In this exercise, the constant \( C \) introduces flexibility into the solution:

• Exponentiating the combined log gives \( |y| = |C_1 x^4| \).
• Here, \( C_1 \) is established as the constant \( e^C \), a transformed representation of \( C \).
• Typically, for simplicity, we often choose \( C_1 \) to be positive, leading to an easier expression, \( y = C_1 x^4 \).

Recognizing the role of the constant of integration allows one to understand how general solutions of differential equations form families of curves, reflecting initial conditions that might be applied later in specific contexts.