Problem 7
Question
State the order of the given ordinary differential equation. Determine whether the equation is linear or nonlinear by matching it with. $$ (\sin \theta) y^{\prime \prime \prime}-(\cos \theta) y^{\prime}=2 $$
Step-by-Step Solution
Verified Answer
The order is 3, and the equation is linear.
1Step 1: Identify the Order of the Differential Equation
The order of a differential equation is determined by the highest derivative present in the equation. In the equation \( (\sin \theta) y^{\prime \prime \prime}-(\cos \theta) y^{\prime}=2 \), the highest derivative is \( y^{\prime \prime \prime} \). Therefore, the order of this differential equation is 3.
2Step 2: Recognize the Form of the Differential Equation
A differential equation is linear if each term is either a constant or a product of a function of independent variables and the dependent variable or its derivatives, without any power or non-linear operations on the dependent variable and its derivatives. For the given equation, \( (\sin \theta) y^{\prime \prime \prime}-(\cos \theta) y^{\prime}=2 \), each term is linear, involving only the first and third derivatives of \( y \), multiplied by functions of \( \theta \).
3Step 3: Conclusion on Linearity
Examining the terms \( (\sin \theta) y^{\prime \prime \prime} \) and \( -(\cos \theta) y^{\prime} \), we see that there are no powers or non-linear functions of \( y \) or its derivatives. Since all terms fit the form of a linear differential equation, this equation is linear.
Key Concepts
Linear Differential EquationsOrder of Differential EquationOrdinary Differential Equation
Linear Differential Equations
Linear differential equations are a specific type of differential equation where the dependent variable and its derivatives appear in a linear fashion. This means each term is either constant or a simple product of a function of independent variables and the dependent variable. Importantly, there should be no powers or other nonlinear operations on the dependent variable and its derivatives.
For an equation to be classified as linear, the structure should be clear:
For an equation to be classified as linear, the structure should be clear:
- No multiplication or division of the dependent variable with its derivatives.
- Each derivative term appears in its first power.
- Any coefficients are functions of independent variables.
Order of Differential Equation
The order of a differential equation is a fundamental concept that helps to define its characteristics and complexity. It is simply the highest derivative of the dependent variable present in the equation.
For example, if the highest derivative is the second derivative, the equation is said to be of second order. In the context of the provided exercise, the highest derivative present is the third derivative (noted as \( y''' \) ). Therefore, the differential equation is of the third order.
Grasping the order of a differential equation is essential as it sets the stage for the type of solution methods that can be applied.
For example, if the highest derivative is the second derivative, the equation is said to be of second order. In the context of the provided exercise, the highest derivative present is the third derivative (noted as \( y''' \) ). Therefore, the differential equation is of the third order.
Grasping the order of a differential equation is essential as it sets the stage for the type of solution methods that can be applied.
Ordinary Differential Equation
An ordinary differential equation (ODE) involves functions of only one independent variable and its derivatives. This distinguishes it from partial differential equations, where multiple independent variables come into play.
In the given example, the equation is an ordinary differential equation because it only involves derivatives with respect to a single variable \( \theta \). This simplicity is key in many real-world scenarios where changes are continuous over a single parameter, like time.
Understanding the nature of ODEs allows for practical applications and effective solving techniques tailored to the problem's conditions.
In the given example, the equation is an ordinary differential equation because it only involves derivatives with respect to a single variable \( \theta \). This simplicity is key in many real-world scenarios where changes are continuous over a single parameter, like time.
Understanding the nature of ODEs allows for practical applications and effective solving techniques tailored to the problem's conditions.
Other exercises in this chapter
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