State whether the given differential equations are linear or nonlinear. Give the order o...

Consider the given differential equation:

$$ (\sin x) y^{-}-(\cos x) y^{\prime}=2 $$

Re-write the given differential equation in the following way:

$$ (\sin x) \frac{d^{3} y}{d x^{3}}-(\cos x) \frac{d y}{d x}-2=0 $$

Organize each term as shown in the table:

\begin{tabular}{|l|l|l|}

\hline Terms & Coefficient & Variable \(y\) and all its derivatives power \\

\hline\((\sin x) \frac{d^{3} y}{d x^{3}}\) & \(\sin x\) & \(\left(\frac{d^{3} y}{d x^{3}}\right)^{1}\), power is 1 \\

\hline\(-(\cos x) \frac{d y}{d x}\) & \(-\cos x\) & \(\left(\frac{d y}{d x}\right)^{\prime}\) power is 1 \\

\hline

\end{tabular}

Check the "second" column here each coefficient depends on only variable \(x\).

And the "third" column here variable \(y\) and all its derivatives are of first degree (having power 1\()\)

Since for a linear equation power must equals to 1 for each term.

Hence the given equation is linear differential equation.

Now to find the order of the given equation look for the order of the derivative in each term, the highest derivative in all represents the order of the differential equation.

\begin{tabular}{|l|l|l|}

\hline Terms & Derivative order & order \\

\hline\((\sin x) \frac{d^{3} y}{d x^{3}}\) & \(\frac{d^{3} y}{d x^{3}}\), third order & 3 \\

\hline\(-(\cos x) \frac{d y}{d x}\) & \(\frac{d y}{d x}\), first order & 1 \\

\hline

\end{tabular}

Clearly it can be concluded from the "third" column the highest order is 3 .

Therefore, the order of the differential equation is 3 .