Chapter 1: Problem 5
Determine the order of the given differential equation; also state whether the equation is linear or nonlinear. $$ \frac{d^{2} y}{d t^{2}}+\sin (t+y)=\sin t $$
Short Answer
Expert verified
Answer: The order of the given differential equation is 2, and it is nonlinear.
Step by step solution
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Order of Differential Equations
When discussing differential equations, the order is a fundamental concept. The order of a differential equation is determined by the highest derivative present in the equation. In simpler terms, it tells us the degree to which a variable is differentiated.
In the example provided, \( \frac{d^{2} y}{d t^{2}} + \sin(t+y) = \sin t \), the highest derivative is \( \frac{d^{2} y}{d t^{2}} \), which is the second derivative of \( y \) with respect to \( t \). Therefore, this equation is of second order. Understanding the order helps categorize differential equations and choose appropriate solution methods.
Always remember:\( n \)-th order differential equations have their highest derivative as the \( n \)-th derivative.
In the example provided, \( \frac{d^{2} y}{d t^{2}} + \sin(t+y) = \sin t \), the highest derivative is \( \frac{d^{2} y}{d t^{2}} \), which is the second derivative of \( y \) with respect to \( t \). Therefore, this equation is of second order. Understanding the order helps categorize differential equations and choose appropriate solution methods.
Always remember:\( n \)-th order differential equations have their highest derivative as the \( n \)-th derivative.
Linear and Nonlinear Equations
Differential equations can be broadly classified as linear or nonlinear based on the presence and interaction of the dependent variable and its derivatives.
A differential equation is linear if it can be expressed in the form:
In contrast, a nonlinear differential equation contains terms where the dependent variable or its derivatives are raised to a power, multiplied together, or are part of nonlinear functions like trigonometric functions.
For the equation \( \frac{d^{2} y}{d t^{2}} + \sin(t+y) = \sin t \), the presence of \( \sin(t+y) \) makes it nonlinear because it involves a combination of \( t \) and \( y \) inside a sine function.
A differential equation is linear if it can be expressed in the form:
- \( a(t)y^{(n)} + b(t)y^{(n-1)} + ... + c(t)y = g(t) \)
- where \( a(t), b(t), c(t) \) are functions of the independent variable \( t \).
In contrast, a nonlinear differential equation contains terms where the dependent variable or its derivatives are raised to a power, multiplied together, or are part of nonlinear functions like trigonometric functions.
For the equation \( \frac{d^{2} y}{d t^{2}} + \sin(t+y) = \sin t \), the presence of \( \sin(t+y) \) makes it nonlinear because it involves a combination of \( t \) and \( y \) inside a sine function.
Second-Order Differential Equations
Second-order differential equations are a specific type of differential equations characterized by the presence of the second derivative. They are prevalent in the study of physical phenomena like oscillations and waves.
These equations often look like \( a(t)\frac{d^{2} y}{d t^{2}} + b(t)\frac{d y}{d t} + c(t)y = g(t) \). The process of solving them can reveal important properties of systems, like stability and behavior over time.
In the exercise example, the equation is \( \frac{d^{2} y}{d t^{2}} + \sin(t+y) = \sin t \), confirming it as a second-order because the highest derivative order is two.
Depending on whether they are linear or nonlinear, solving second-order equations can vary in difficulty and methods applicable. Techniques for solving second-order differential equations include methods like undetermined coefficients, variation of parameters, and numerical methods for more complex cases.
These equations often look like \( a(t)\frac{d^{2} y}{d t^{2}} + b(t)\frac{d y}{d t} + c(t)y = g(t) \). The process of solving them can reveal important properties of systems, like stability and behavior over time.
In the exercise example, the equation is \( \frac{d^{2} y}{d t^{2}} + \sin(t+y) = \sin t \), confirming it as a second-order because the highest derivative order is two.
Depending on whether they are linear or nonlinear, solving second-order equations can vary in difficulty and methods applicable. Techniques for solving second-order differential equations include methods like undetermined coefficients, variation of parameters, and numerical methods for more complex cases.
