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 $\mathit{x}\frac{{\mathbf{d}}^{\mathbf{3}}\mathbf{y}}{\mathbf{d}{\mathbf{x}}^{\mathbf{3}}}\mathbf{-}{\left(\frac{dy}{dx}\right)}^{\mathbf{4}}\mathbf{+}\mathit{y}\mathbf{=}\mathbf{0}\mathbf{}\left(6\right)$.

Short answer

Answer:

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

Step-by-step solution

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

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

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

Determine whether it is linear or nonlinear and state the order

If the given equation is linear, then the term$\frac{dy}{dx}$must have the power of 1, but the term$\frac{dy}{dx}$is to the power of 4.Moreover, by the classification of linearity, the given differential equation cannot be compared to the form

$a_3\left(x\right)y"\text{'}+a_{2}x\left(y\right)"+a_1\left(x\right)y\text{'}+a_0\left(x\right)y=g\left(x\right)$. So, the given equation is nonlinear.

The highest derivative present in the differential equation is 3, as $n=3$. So, the given differential equation is of the third order.