Question

Consider the Ordinary Differential Equation (ODE)

d²y/dt +2(dy/dt) +sin(y)=0
where y is the dependent variable which is a function of the independent variable t. Select the terms that accurately describe this ODE (there are 3 correct answers out of 5).
A. Linear
B. Autonomous
C. Cauchy-Euler
D. Homogeneous
E. Second Order

Answer 1

The correct descriptions for the ODE d²y/dt² + 2(dy/dt) + sin(y) = 0 are Autonomous, Homogeneous, and Second Order, because it does not explicitly contain the independent variable, all terms are functions of the dependent variable or its derivatives, and it has a second derivative, respectively.

Step-by-step explanation:

The Ordinary Differential Equation (ODE) in question is d²y/dt² + 2(dy/dt) + sin(y) = 0. Now, let's break down the properties of this ODE to select the correct descriptions.

• The ODE is non-linear because of the sine function, which is a non-linear function of the dependent variable, y.
• It is autonomous because the equation does not explicitly contain the independent variable, t.
• The term Cauchy-Euler does not apply here because a Cauchy-Euler equation has variable coefficients that are powers of the independent variable, which this ODE does not have.
• This ODE is homogeneous because all terms are a function of y or its derivatives, and there are no separate terms that are just functions of t (i.e., there is no 'forcing' term that is independent of y).
• It is indeed a second-order ODE since the highest derivative is the second derivative of y with respect to t.

Therefore, the correct answers describing the given ODE are B. Autonomous, D. Homogeneous, and E. Second Order.