Problem 1
Question
State the order of the given ordinary differential equation. Determine whether the equation is linear or nonlinear by matching it with. $$ (1-x) y^{\prime \prime}-4 x y^{\prime}+5 y=\cos x $$
Step-by-Step Solution
Verified Answer
The equation is a second-order linear differential equation.
1Step 1: Identify the Highest Derivative
The given differential equation is \((1-x) y^{\prime\prime} - 4x y^{\prime} + 5y = \cos x\). First, identify the highest derivative in the equation. The highest derivative here is \(y^{\prime\prime}\), which is the second derivative of \(y\).
2Step 2: Determine the Order of the Equation
The order of a differential equation is determined by the highest derivative present in the equation. Since the highest derivative in this equation is \(y^{\prime\prime}\), the equation is of second order.
3Step 3: Analyze Linearity by Structure
To determine if the equation is linear, check if all terms are either a constant or involve the dependent variable \(y\) or its derivatives linearly (i.e., no powers or products of \(y\), \(y'\), \(y''\), etc.). The given equation, \((1-x) y^{\prime\prime} - 4x y^{\prime} + 5y = \cos x\), features no such nonlinear terms. Thus, it is a linear equation.
Key Concepts
Second-Order Differential EquationLinearity of Differential EquationsDifferential Equation Order Determination
Second-Order Differential Equation
Differential equations are divided into orders based on the highest derivative present. The order is crucial for understanding the nature of a differential equation, influencing the solution methods and the complexity of solutions.
In the example \[(1-x) y^{\prime\prime} - 4x y^{\prime} + 5y = \cos x\]
the highest derivative is \( y^{\prime\prime} \). This signifies that the given differential equation is of second order.
In practical terms, higher-order differential equations typically model more complex systems, often representing real-world phenomena with multiple changing rates, such as acceleration in physics. Remember, the order of a differential equation helps you decide on the approach and techniques needed for finding its solutions, impacting which mathematical methods to employ.
In the example \[(1-x) y^{\prime\prime} - 4x y^{\prime} + 5y = \cos x\]
the highest derivative is \( y^{\prime\prime} \). This signifies that the given differential equation is of second order.
In practical terms, higher-order differential equations typically model more complex systems, often representing real-world phenomena with multiple changing rates, such as acceleration in physics. Remember, the order of a differential equation helps you decide on the approach and techniques needed for finding its solutions, impacting which mathematical methods to employ.
Linearity of Differential Equations
Linearity in differential equations is about maintaining certain mathematical relationships. A linear differential equation means that all derivatives and the dependent variable, usually denoted as \( y \), appear to the power of 1, without being multiplied by each other.
For the equation given: \[(1-x) y^{\prime\prime} - 4x y^{\prime} + 5y = \cos x\]
each term on the left involves either a derivative of \( y \) or \( y \) itself, and all are linear.
No term in the equation has \( y \) or its derivatives raised to a power greater than 1, nor do they multiply each other. Also, there are no functions of \( y \) (such as \( \sin(y) \) or \( e^y \)) in the equation. This confirms the equation is linear, adhering to the rules for linearity, which makes it easier to solve using standard techniques.
For the equation given: \[(1-x) y^{\prime\prime} - 4x y^{\prime} + 5y = \cos x\]
each term on the left involves either a derivative of \( y \) or \( y \) itself, and all are linear.
- The term \((1-x)y^{\prime\prime}\) is linear in its highest derivative \(y^{\prime\prime}\).
- The term \(-4xy^{\prime}\) involves \(y^{\prime}\) linearly.
- The term \(5y\) involves \(y\) linearly.
No term in the equation has \( y \) or its derivatives raised to a power greater than 1, nor do they multiply each other. Also, there are no functions of \( y \) (such as \( \sin(y) \) or \( e^y \)) in the equation. This confirms the equation is linear, adhering to the rules for linearity, which makes it easier to solve using standard techniques.
Differential Equation Order Determination
Identifying the order of a differential equation is one of the first steps in solving it. Finding the highest derivative provides essential clues about how the equation can be approached.
The order affects decisions about which solution methods to use, because specific techniques are developed for solving equations of certain orders.
Consider the given example: \[(1-x) y^{\prime\prime} - 4x y^{\prime} + 5y = \cos x\]
The presence of \( y^{\prime\prime} \) as the highest derivative indicates a second-order differential equation.
Accurately determining the order is foundational, as it lays the groundwork for further analysis and application of proper solving techniques like separation of variables, integrating factors, or use of characteristic equations.
The order affects decisions about which solution methods to use, because specific techniques are developed for solving equations of certain orders.
Consider the given example: \[(1-x) y^{\prime\prime} - 4x y^{\prime} + 5y = \cos x\]
The presence of \( y^{\prime\prime} \) as the highest derivative indicates a second-order differential equation.
- A first-order differential equation would involve no higher derivative than \( y^{\prime} \).
- Third-order, fourth-order, and so forth, would involve derivatives such as \( y^{\prime\prime\prime} \) and \( y^{\prime\prime\prime\prime} \).
Accurately determining the order is foundational, as it lays the groundwork for further analysis and application of proper solving techniques like separation of variables, integrating factors, or use of characteristic equations.
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