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 $\left(1-x\right)\mathit{y}\mathbf{"}\mathbf{-}\mathbf{4}\mathit{x}\mathit{y}\mathbf{\text{'}}\mathbf{+}\mathbf{5}\mathit{y}\mathbf{=}\mathit{c}\mathit{o}\mathit{s}\mathbf{}\mathit{x}$.

Short answer

Answer:

The differential equation is of the second 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 mathematically presented as

${\mathbf{a}}_{\mathbf{n}}\left(x\right)\frac{{\mathbf{d}}^{\mathbf{n}}\mathbf{y}}{{\mathbf{dx}}^{\mathbf{n}}}\mathbf{+}{\mathbf{a}}_{\mathbf{n}\mathbf{-}\mathbf{1}}\left(x\right)\frac{{\mathbf{d}}^{\mathbf{n}\mathbf{-}\mathbf{1}}\mathbf{y}}{{\mathbf{dx}}^{\mathbf{n}\mathbf{-}\mathbf{1}}}\mathbf{+}\mathbf{L}\mathbf{+}{\mathbf{a}}_{\mathbf{1}}\left(x\right)\frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{+}{\mathbf{a}}_{\mathbf{0}}\left(x\right)\mathbf{y}\mathbf{=}\mathbf{g}\left(x\right)$.

Determine whether it is linear or nonlinear and state the order

By the classification of linearity, the given differential equation matches the form$a_2\left(x\right)y"+a_1\left(x\right)y\text{'}+a_0\left(x\right)y=g\left(x\right)$. On comparing the given equation with this form, we obtain the parameters equal to,

$$\begin{gathered} a_2=\left(1-x\right) \\ {a}_1=-4x \\ {a}_0=5 \\ g\left(x\right)=\cos x \end{gathered}$$

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

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