91Ó°ÊÓ

In Problems $\mathbf{15}-\mathbf{24}$ , solve for $\mathbf{Y}\mathbf{\left(}\mathbf{s}\mathbf{\right)}$, the Laplace transform of the solution $\mathbf{y}\left(\mathbf{t}\right)$to the given initial value problem.

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{3}\mathbf{y}\mathbf{=}{\mathbf{t}}^{\mathbf{3}}\mathbf{;}\mathbf{y}\left(\mathbf{0}\right)\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{y}\mathbf{\text{'}}\left(\mathbf{0}\right)\mathbf{=}\mathbf{0}$

Use Heaviside's expansion formula derived in Problem 40 to determine the inverse Laplace transform of

$F\left(s\right)=\frac{3s^2-16s+5}{(s+1)(s-3)(s-2)}$

Find a general solution for the differential equation with x as the independent variable:

$y\text{'}\text{'}\text{'}+3y\text{'}\text{'}−4y\text{'}−6y=0$

Question 10: In Problems, find the power series expansion for f(x)+g(x), given the expansions for f(x) and g(x).

10.

In Problems 9-13, determine whether the given relation is an implicit solution to the given differential equation. Assume that the relationship implicitly defines y as a function of x and use implicit differentiation.

$\mathbf{y}\mathbf{-}{\mathbf{log}}_{\mathbf{e}}\mathbf{y}\mathbf{=}{\mathbf{x}}^{\mathbf{2}}\mathbf{+}\mathbf{1}$,$\frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{=}\frac{\mathbf{2}\mathbf{xy}}{\mathbf{y}\mathbf{-}\mathbf{1}}$

Decide whether the statement made is True or False. The relation $\mathbf{sin}\mathbf{y}\mathbf{+}{\mathbf{e}}^{\mathbf{y}}\mathbf{=}{\mathbf{x}}^{\mathbf{6}}\mathbf{-}{\mathbf{x}}^{\mathbf{2}}\mathbf{+}\mathbf{x}\mathbf{+}\mathbf{1}$ is an implicit solution to $\frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{=}\frac{\mathbf{6}{\mathbf{x}}^{\mathbf{5}}\mathbf{-}\mathbf{2}\mathbf{x}\mathbf{+}\mathbf{1}}{\mathbf{cos}\mathbf{y}\mathbf{+}{\mathbf{e}}^{\mathbf{y}}}$.

Question 11: In Problem, find the first three nonzero terms in the power series expansion for the product f(x) g(x).

In Problems 9–13, determine whether the given relation is an implicit solution to the given differential equation. Assume that the relationship implicitly defines y as a function of x and use implicit differentiation.

${\mathbf{e}}^{\mathbf{xy}}\mathbf{+}\mathbf{y}\mathbf{=}\mathbf{x}\mathbf{-}\mathbf{1}$,$\frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{=}\frac{{\mathbf{e}}^{\mathbf{-}\mathbf{xy}}\mathbf{-}\mathbf{y}}{{\mathbf{e}}^{\mathbf{-}\mathbf{xy}}\mathbf{+}\mathbf{x}}$

The initial value problem \[\frac{{{\bf{dx}}}}{{{\bf{dt}}}}{\bf{ = 3}}{{\bf{x}}^{\frac{{\bf{2}}}{{\bf{3}}}}}{\bf{,}}\;{\bf{x(0) = 1}}\] has a unique solution in some open interval around t = 0.

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

If it is, then solve it.

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