91Ó°ÊÓ

Determine the form of a particular solution for the differential equation. (Do not evaluate coefficients.) $\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{6}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{9}\mathbf{y}\mathbf{=}\mathbf{5}{\mathbf{t}}^6{\mathbf{e}}^{3t}$

Let ${\mathbf{y}}_{\mathbf{1}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}{\mathbf{t}}^{\mathbf{2}}$and${\mathbf{y}}_{\mathbf{2}}\left(\mathbf{t}\right)\mathbf{=}\mathbf{2}\mathbf{t}\left|\mathbf{t}\right|$. Are${\mathbf{y}}_{\mathbf{1}}$and${\mathbf{y}}_2$linearly independent on the interval

(a). $\mathbf{[}\mathbf{0}\mathbf{,}\mathbf{∞}\mathbf{)}$

(b). $\mathbf{(}\mathbf{-}\mathbf{∞}\mathbf{,}\mathbf{0}\mathbf{]}$

(c). $\mathbf{(}\mathbf{-}\mathbf{∞}\mathbf{,}\mathbf{∞}\mathbf{)}$

(d). Compute the Wronskian $\mathbf{W}\left[{\mathbf{y}}_{\mathbf{1}}\mathbf{,}{\mathbf{y}}_{\mathbf{2}}\right]\mathbf{\left(}\mathbf{t}\mathbf{\right)}$on the interval $\mathbf{(}\mathbf{-}\mathbf{∞}\mathbf{,}\mathbf{∞}\mathbf{)}$.

Find a general solution to the given differential equation.

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{=}\mathbf{5}{\mathbf{x}}^{\mathbf{-}\mathbf{1}}\mathbf{y}\mathbf{\text{'}}\mathbf{-}\mathbf{13}{\mathbf{x}}^{\mathbf{-}\mathbf{2}}\mathbf{y}\mathbf{,}     \mathbf{x}\mathbf{>}\mathbf{0}$

Find the solution to the initial value problem.$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\left(\mathbf{θ}\right)\mathbf{-}\mathbf{y}\left(\mathbf{θ}\right)\mathbf{=}\mathbf{²õ¾\pm ²Ôθ}\mathbf{-}{\mathbf{e}}^{2\mathrm{θ}}\mathbf{;}\text{ â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰}\mathbf{y}\left(\mathbf{0}\right)\mathbf{=}\mathbf{1}\mathbf{,}\text{ â¶Ä‰â¶Ä‰â¶Ä‰}\mathbf{y}\mathbf{\text{'}}\left(\mathbf{0}\right)\mathbf{=}\mathbf{-}\mathbf{1}$

In Problems 27–32, use Definition 1 to determine whether the functions y₁and y₂are linearly dependent on the interval (0, 1).

29. y₁(t) = te^2t, y₂(t) = e^2t

Determine the form of a particular solution for the differential equation. (Do not evaluate coefficients.)

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{3}\mathbf{y}\mathbf{\text{'}}\mathbf{-}\mathbf{7}\mathbf{y}\mathbf{=}{\mathbf{t}}^4{\mathbf{e}}^t$

Find a general solution to the following higher-order equations.

(a)$\mathit{y}\text{'}\text{'}\text{'}\mathbf{-}\mathit{y}\text{'}\text{'}\mathbf{+}\mathit{y}\text{'}\mathbf{+}\mathbf{3}\mathit{y}\mathbf{=}\mathbf{0}$

(b)$\mathit{y}\text{'}\text{'}\text{'}\mathbf{+}\mathbf{2}\mathit{y}\text{'}\text{'}\mathbf{+}\mathbf{5}\mathit{y}\text{'}\mathbf{-}\mathbf{26}\mathit{y}\mathbf{=}\mathbf{0}$

(c)${\mathit{y}}^{iv}\mathbf{+}\mathbf{13}\mathit{y}\text{'}\text{'}\mathbf{+}\mathbf{36}\mathit{y}\mathbf{=}\mathbf{0}$

Prove that if ${\mathbf{y}}_{\mathbf{1}}$and${\mathbf{y}}_{\mathbf{2}}$are linearly independent solutions of$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{py}\mathbf{\text{'}}\mathbf{+}\mathbf{qy}\mathbf{=}\mathbf{0}$on$(\mathbf{a}\mathbf{,}\mathbf{b})$, then they cannot both be zero at the same point${\mathbf{t}}_{\mathbf{0}}$in$\left(a,b\right)$

Find the solution to the given initial value problem.

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{4}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{7}\mathbf{y}\mathbf{=}\mathbf{0}\mathbf{;}     \mathbf{y}\left(\mathbf{0}\right)\mathbf{=}\mathbf{1}\mathbf{,}    \mathbf{y}\mathbf{\text{'}}\left(\mathbf{0}\right)\mathbf{=}\mathbf{-}\mathbf{2}$

In Problems 1–8, find a general solution to the differential equation using the method of variation of parameters. $\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{y}\mathbf{=}\mathbf{sect}$