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Chapter 1: Problem 7

State the order of the given ordinary differential equation. Determine whether the equation is linear or nonlinear by matching it with. $$ (\sin \theta) y^{\prime \prime \prime}-(\cos \theta) y^{\prime}=2 $$

Short Answer

Expert verified
The order is 3, and the equation is linear.

Step by step solution

01

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.
02

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 \( \theta \).
03

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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:
  • 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.
Understanding these characteristics helps in identifying and solving linear differential equations more effectively.
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.
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.

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Most popular questions from this chapter

Verify that the piecewise-defined function $$ y=\left\\{\begin{array}{ll} -x^{2}, & x<0 \\ x^{2}, & x \geq 0 \end{array}\right. $$ is a solution of the differential equation \(x y^{\prime}-2 y=0\) on the interval \((-\infty, \infty)\)

(a) Verify that \(y=\tan (x+c)\) is a one-parameter family of solutions of the differential equation \(y^{\prime}=1+y^{2}\). (b) Since \(f(x, y)=1+y^{2}\) and \(\partial f / \partial y=2 y\) are continuous everywhere, the region \(R\) in Theorem \(1.2 .1\) can be taken to be the entire \(x y\) -plane. Use the family of solutions in part (a) to find an explicit solution of the first-order initialvalue problem \(y^{\prime}=1+y^{2}, y(0)=0 .\) Even though \(x_{0}=0\) is in the interval \((-2,2)\), explain why the solution is not defined on this interval. (c) Determine the largest interval \(I\) of definition for the solution of the initial-value problem in part (b).

In Problems 37 and 38 , verify that the indicated pair of functions is a solution of the given system of differential equations on the interval \((-\infty, \infty)\) $$ \begin{aligned} &\frac{d^{2} x}{d t^{2}}=4 y+e^{t} \\ &\frac{d^{2} y}{d t^{2}}=4 x-e^{t} \\ &x=\cos 2 t+\sin 2 t+\frac{1}{5} e^{t} \\ &y=-\cos 2 t-\sin 2 t-\frac{1}{5} e^{t} \end{aligned} $$

(a) Byinspection, find a one-parameter family of solutions of the differential equation \(x y^{\prime}=y .\) Verify that each member of the family is a solution of the initial-value problem \(x y^{\prime}=y, y(0)=0\) (b) Explain part (a) by determining a region \(R\) in the \(x y\) -plane for which the differential equation \(x y^{\prime}=y\) would have a unique solution through a point \(\left(x_{0}, y_{0}\right)\) in \(R\). (c) Verify that the picecwise-defined function $$ y=\left\\{\begin{array}{ll} 0, & x<0 \\ x, & x \geq 0 \end{array}\right. $$ satisfies the condition \(y(0)=0\). Determine whether this function is also a solution of the initial-value problem in part (a).

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