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 ${\mathit{t}}^{\mathbf{5}}{\mathit{y}}^{\left(4\right)}\mathbf{-}{\mathit{t}}^{\mathbf{3}}\mathit{y}\mathbf{"}\mathbf{+}\mathbf{6}\mathit{y}\mathbf{=}\mathbf{0}$.

Short answer

Answer:

The differential equation is of the fourth order, and the equation is 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 given mathematically presented as ${\mathit{a}}_{\mathbf{n}}\left(x\right)\frac{{\mathbf{d}}^{\mathbf{n}}\mathbf{y}}{\mathbf{d}{\mathbf{x}}^{\mathbf{n}}}\mathbf{+}{\mathit{a}}_{\mathbf{n}\mathbf{-}\mathbf{1}}\left(x\right)\frac{{\mathbf{d}}^{\mathbf{n}\mathbf{-}\mathbf{1}}\mathbf{y}}{\mathbf{d}{\mathbf{x}}^{\mathbf{n}\mathbf{-}\mathbf{1}}}\mathbf{+}\mathit{L}\mathbf{+}{\mathit{a}}_{\mathbf{1}}\left(x\right)\frac{\mathbf{d}\mathbf{y}}{\mathbf{d}\mathbf{x}}\mathbf{+}{\mathit{a}}_{\mathbf{0}}\left(x\right)\mathit{y}\mathbf{=}\mathit{g}\left(x\right)$.

Determine whether it is linear or nonlinear and state the order

Bby the classification of linearity, the given differential equation matches the form. On comparing the given equation with this form, we obtain the parameters equal to,

$$\begin{gathered} a_4=t^5 \\ {a}_3=0 \\ {a}_2=-t^3 \end{gathered}$$and $$\begin{gathered} a_1=0 \\ {a}_0=6 \\ g\left(t\right)=0 \end{gathered}$$

The parameter is linear in its derivative, so, the given differential equation is linear.

The highest derivative present in the differential equation is 4, as $n=4$. So, the given differential equation is of the fourth order.