91影视

In Problems 9鈥�20, determine whether the equation is exact.

If it is, then solve it.

$\left[\mathbf{2}\mathbf{x}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}\mathbf{-}\mathbf{cos}\mathbf{(}\mathbf{x}\mathbf{+}\mathbf{y}\mathbf{)}\right]\mathbf{dx}\mathbf{+}\left[\mathbf{2}\mathbf{xy}\mathbf{-}\mathbf{cos}\mathbf{(}\mathbf{x}\mathbf{+}\mathbf{y}\mathbf{)}\mathbf{-}{\mathbf{e}}^{\mathbf{y}}\right]\mathbf{dy}\mathbf{=}\mathbf{0}$

If $\mathbf{xM}\left(\mathbf{x}\mathbf{,}\mathbf{y}\right)\mathbf{+}\mathbf{yN}\left(\mathbf{x}\mathbf{,}\mathbf{y}\right)鈮�\mathbf{0}$, find the solution to the equation $\mathbf{M}\left(\mathbf{x}\mathbf{,}\mathbf{y}\right)\mathbf{dx}\mathbf{+}\mathbf{N}\left(\mathbf{x}\mathbf{,}\mathbf{y}\right)\mathbf{dy}\mathbf{=}\mathbf{0}$.

Question: In Problems 1-30, solve the equation.

$\frac{\mathrm{dy}}{\mathrm{dx}}\mathbf{=}{\left(2x+y-1\right)}^2$

Use the method discussed under 鈥淓quations of the Form $\frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{=}\mathbf{G}\left(\mathbf{ax}\mathbf{+}\mathbf{by}\right)$鈥� to solve problems17-20.

$\frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{=}{\left(\mathbf{x}\mathbf{-}\mathbf{y}\mathbf{+}\mathbf{5}\right)}^{\mathbf{2}}$

Fluid Flow. The streamlines associated with a certain fluid flow are represented by the family of curves $\mathbf{y}\mathbf{=}\mathbf{x}\mathbf{-}\mathbf{1}\mathbf{+}{\mathbf{ke}}^{\mathbf{-}\mathbf{x}}$. The velocity potentials of the flow are just the orthogonal trajectories of this family.

[Hint: First express the family$\mathbf{y}\mathbf{=}\mathbf{x}\mathbf{-}\mathbf{1}\mathbf{+}{\mathbf{ke}}^{\mathbf{-}\mathbf{x}}$ in the form $\mathbf{F}\left(\mathbf{x}\mathbf{,}\mathbf{y}\right)\mathbf{=}\mathbf{k}$.]

Question: In Problems 1-30, solve the equation.

$\left({\mathbf{x}}^2\mathbf{-}\mathbf{3}{\mathbf{y}}^2\right)\mathbf{dx}\mathbf{+}\mathbf{2}\mathbf{xy}\text{鈥�}\mathbf{dy}\mathbf{=}\mathbf{0}$

In problem $\mathbf{1}\mathbf{-}\mathbf{6}$, determine whether the differential equation is separable $\frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{-}\mathit{s}\mathit{i}\mathit{n}\mathbf{(}\mathbf{x}\mathbf{+}\mathbf{y}\mathbf{)}\mathbf{=}\mathbf{0}$.

In problems Classify the equation as separable, linear, exact, or none of these. Notice that some equations may have more than one classification.

$\mathbf{(}{\mathbf{x}}^{\mathbf{2}}\mathbf{y}\mathbf{+}{\mathbf{x}}^{\mathbf{4}}\mathbf{cos}\mathbf{x}\mathbf{)}\mathbf{dx}\mathbf{-}{\mathbf{x}}^{\mathbf{3}}\mathbf{dy}\mathbf{=}\mathbf{0}$

In Problems \(1 - 6\), determine whether the given equation is separable, linear, neither, or both.

\({{\bf{x}}^{\bf{2}}}\frac{{{\bf{dy}}}}{{{\bf{dx}}}}{\bf{ + sinx - y = 0}}\).

Question: In Problems 1 - 30, solve the equation.

$\text{鈥�}\frac{\mathrm{dy}}{\mathrm{dx}}\mathbf{=}\frac{{\mathbf{e}}^{x+y}}{y-1}$