Question

In Problems$\mathbf{1}\mathbf{-}\mathbf{8}$state the order of the given ordinary differential equation. Determine whether the equation is linear or nonlinear by matching it with $$\begin{gathered} \frac{{\mathbf{d}}^{\mathbf{2}}\mathbf{u}}{\mathbf{d}{\mathbf{r}}^{\mathbf{2}}}\mathbf{+}\frac{\mathbf{d}\mathbf{u}}{\mathbf{d}\mathbf{r}}\mathbf{+}\mathit{u}\mathbf{=}\mathit{c}\mathit{o}\mathit{s}\left(r+u\right) \\ \left(6\right) \end{gathered}$$.

Short answer

Answer:

The differential equation is of the second order, and the equation is non-linear.

Step-by-step solution

Step by Step solution: Step 1: Classification of linearity.

If is linear in , then the order ordinary differential equation is said to be linear. The form of the equation is mathematically presented as

$a_n\left(x\right)\frac{d^{n}y}{d{x}^n}+a_{n-1}\left(x\right)\frac{d^{n-1}y}{dxn-1}+L+a_1\left(x\right)\frac{dy}{dx}+a_0\left(x\right)y=g\left(x\right)$.

Determine whether it is linear or nonlinear and state the order

By the classification of linearity, the given differential equation should be in the form$a_2\left(r\right)\frac{d^{2}u}{d{r}^2}+a_1\left(r\right)\frac{du}{dr}+a_0\left(r\right)u=g\left(r\right)$. Thus, the term must be dependent only on but the term has dependence both on u and r. So, the given differential equation is nonlinear.

The highest derivative present in the differential equation is 2, as $n=2$. So, the given differential equation is of the second order.