Questions: Determine the order of the given differential equation and state whether the equation is linear or nonlinear. d^4 u/d r^4 + d u/d r + 5 u = cos (r+u) (a) The order of this differential equation is (b) The equation is

Determine the order of the given differential equation and state whether the equation is linear or nonlinear. d^4 u/d r^4 + d u/d r + 5 u = cos (r+u) (a) The order of this differential equation is (b) The equation is
Transcript text: Determine the order of the given differential equation and state whether the equation is linear or nonlinear. \[ \frac{d^{4} u}{d r^{4}}+\frac{d u}{d r}+5 u=\cos (r+u) \] (a) The order of this differential equation is $\square$ (b) The equation is $\square$ Choose
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Solution

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Determine the order of the given differential equation.

Identify the highest derivative in the equation.

The highest derivative in the equation is \(\frac{d^{4} u}{d r^{4}}\), which is a fourth-order derivative.

\(\boxed{\text{The order of this differential equation is } 4}\)

State whether the equation is linear or nonlinear.

Check for linearity by examining the terms involving \(u\) and its derivatives.

The term \(\cos(r + u)\) involves \(u\) in a nonlinear way (inside a trigonometric function), making the equation nonlinear.

\(\boxed{\text{The equation is nonlinear}}\)

\(\boxed{\text{The order of this differential equation is } 4}\)
\(\boxed{\text{The equation is nonlinear}}\)

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