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Determine the order of the given differential equation and state whether the equation is linear
or nonlinear.
(a) The order of this differential equation is
(b) The equation is Choose

Ask by Evans Park. in the United States
Mar 23,2025

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(a) The order of the differential equation is 2.
(b) The equation is linear.

Solución

To analyze the given differential equation
we will determine the order and whether it is linear or nonlinear.

(a) Order of the Differential Equation

The order of a differential equation is defined as the highest derivative present in the equation. In this case, we have:
  • (the second derivative of )
  • (the first derivative of )
  • (the function itself)
The highest derivative is , which is the second derivative. Therefore, the order of this differential equation is 2.

(b) Linearity of the Equation

A differential equation is considered linear if it can be expressed in the form:
where are functions of only, and is also a function of . Additionally, the dependent variable and its derivatives must appear to the first power and must not be multiplied together.
In the given equation:
  • The coefficients , , and are functions of .
  • The dependent variable and its derivatives and appear to the first power and are not multiplied together.
Thus, the equation is linear.

Summary

(a) The order of this differential equation is 2.
(b) The equation is linear.

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The Deep Dive

The order of this differential equation is 2 because the highest derivative present is , which represents the second derivative of . This indicates that the behavior of the function relies on its second derivative, making it a second-order differential equation.
The equation is nonlinear since it contains a term involving the product of the function and its derivatives, specifically and . Nonlinear equations exhibit complexity in their behavior, often leading to a rich variety of solutions compared to linear ones.

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