Problem 33 - Advanced Engineering Mathematics

Question

Use the concept that \(y=c,-\infty

Step-by-Step Solution

Answer

The differential equation has a constant solution \(y = 2\).

Step 1: Rewrite the Differential Equation

The given differential equation is \(3x \frac{dy}{dx} + 5y = 10\). Our objective is to determine if this equation has constant solutions.

Step 2: Identify the Condition for Constant Solutions

Constant solutions mean \(y = c\), where \(c\) is a constant. For a constant solution, the derivative \(y'\) should be zero, i.e., \(\frac{dy}{dx} = 0\).

Step 3: Substitute Constant Solution Condition

Substitute \(\frac{dy}{dx} = 0\) into the equation. This gives us: \(3x(0) + 5y = 10\). Simplifying, we get \(5y = 10\).

Step 4: Solve for Constant y

Solving \(5y = 10\), we find \(y = \frac{10}{5} = 2\). This means if \(y = 2\), then \(y' = 0\) satisfies the differential equation.

Step 5: Conclusion on Constant Solutions

Therefore, \(y = 2\) is indeed a constant solution of the differential equation \(3x \frac{dy}{dx} + 5y = 10\).

Key Concepts

Constant Functions

The concept of constant functions forms a foundational element in calculus and mathematical analysis. A constant function is a function where the output value remains the same, regardless of the input. In simpler terms:

• The function could be written as: \(y = c\), where \(c\) is a constant number like 2, 3, or 5.
• Graphically, a constant function is represented by a horizontal line on the graph, indicating no change as the input \(x\) values change.
• This function helps in modeling situations where a uniform rate is expected.

Constant functions are crucial in understanding and simplifying complex mathematical equations, especially as they relate to real-world scenarios where conditions might remain unchanged.

Constant Solutions

In the realm of differential equations, finding constant solutions is a typical task. A constant solution suggests that the solution to the differential equation does not change over time or input variations.

• A constant solution will take the form \(y = c\), where \(c\) is a fixed number.
• For a differential equation to have a constant solution, the derivative of the function \(y'\) should equal zero.
• For example, consider the equation given: \(3x \frac{dy}{dx} + 5y = 10\).

By substituting \(\frac{dy}{dx} = 0\), we simplify the equation and solve for \(y\). In our case, it results in \(y = 2\), meaning this is a constant solution to the equation. It's important to find these solutions to identify stable states in dynamic systems.

Derivatives

Derivatives are fundamental in calculus, representing the rate of change of a function with respect to its variable. This concept is essential to understand how variables in equations change and interact.

• A derivative is often denoted as \(\frac{dy}{dx}\) or \(y'\).
• If \(y' = 0\), it indicates no change in \(y\) as \(x\) changes, suggestive of a constant function.
• Derivatives also enable the determination of increasing or decreasing intervals, concavity, and points of inflection in functions.

In solving the differential equation \(3x \frac{dy}{dx} + 5y = 10\) for constant solutions, we begin by setting \(y' = 0\), which simplifies our task to solving a simple algebraic equation. Derivatives are a key tool in analyzing and solving mathematical problems across various fields.

Mathematical Equations

Mathematical equations are expressions that assert equality between two quantities and involve variables, constants, and operators. In solving real-world problems, understanding the roles of variables and constants is crucial.

• Equations can represent simple relationships like \(5y = 10\), as in our example, or complex systems in science and engineering.
• Differential equations are a type of mathematical equation involving derivatives that depict how things change over time.
• Our original equation, \(3x \frac{dy}{dx} + 5y = 10\), combines these elements to model dynamic behavior.

Solving these equations often involves determining the values of variables that satisfy the equation, helping to predict and understand variable relationships under different conditions. They are foundational in fields like physics, economics, and engineering.