91Ó°ÊÓ

Find a particular solution to the differential equation.

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{2}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{2}\mathbf{y}\mathbf{=}\mathbf{4}{\mathbf{te}}^{-t}\mathbf{cost}$

Find the solution to the initial value problem.

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\left(\mathbf{x}\right)\mathbf{-}\mathbf{y}\mathbf{\text{'}}\left(\mathbf{x}\right)\mathbf{-}\mathbf{2}\mathbf{y}\left(\mathbf{x}\right)\mathbf{=}\mathbf{cosx}\mathbf{-}\mathbf{sin}\mathbf{2}\mathbf{x}\mathbf{;}\text{ â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰}\mathbf{y}\left(\mathbf{0}\right)\mathbf{=}\frac{-7}{20}\mathbf{,}\text{ â¶Ä‰â¶Ä‰â¶Ä‰}\mathbf{y}\mathbf{\text{'}}\left(\mathbf{0}\right)\mathbf{=}\frac{1}{5}$

Determine the form of a particular solution for the differential equation. (Do not evaluate coefficients.) $\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{9}\mathbf{y}\mathbf{=}\mathbf{4}{\mathbf{t}}^3\mathbf{sin}\mathbf{3}\mathbf{t}$

Solve the given initial value problem. $\mathbf{y}\text{'}\text{'}\text{'}\mathbf{-}\mathbf{4}\mathbf{y}\text{'}\text{'}\mathbf{+}\mathbf{7}\mathbf{y}\text{'}\mathbf{-}\mathbf{6}\mathbf{y}\mathbf{=}\mathbf{0}\mathbf{;}$$\mathbf{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}1\mathbf{,}\mathbf{y}\text{'}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{,}$$\mathbf{y}\text{'}\text{'}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}$

Consider the linear equation ${\mathbf{t}}^{\mathbf{2}}\mathbf{y}\text{'}\text{'}\mathbf{-}\mathbf{3}\mathbf{ty}\text{'}\mathbf{+}\mathbf{3}\mathbf{y}\mathbf{=}\mathbf{0}$for$\mathbf{-}\mathbf{∞}\mathbf{<}\mathbf{t}\mathbf{<}\mathbf{∞}$

(a). Verify that ${\mathbf{y}}_{\mathbf{1}}\left(\mathbf{t}\right)\mathbf{=}\mathbf{t}$and ${\mathbf{y}}_{\mathbf{2}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}{\mathbf{t}}^{\mathbf{3}}$are two solutions to 21on $\mathbf{(}\mathbf{-}\mathbf{∞}\mathbf{,}\mathbf{∞}\mathbf{)}$. Furthermore, show that ${\mathbf{y}}_{\mathbf{1}}\left({\mathbf{t}}_{\mathbf{0}}\right){\mathbf{y}}_{\mathbf{2}}^{}\text{'}\left({\mathbf{t}}_{\mathbf{0}}\right){\mathbf{-}\mathbf{y}}_{\mathbf{1}}^{}\text{'}\left({\mathbf{t}}_{\mathbf{0}}\right){\mathbf{y}}_{\mathbf{2}}\left({\mathbf{t}}_{\mathbf{0}}\right)≠\mathbf{0}\mathbf{,}{\mathbf{t}}_{\mathbf{0}}\mathbf{=}\mathbf{1}$.

(b). Prove that ${\mathbf{y}}_1\left(\mathbf{t}\right)$and ${\mathbf{y}}_{\mathbf{2}}\left(\mathbf{t}\right)$are linearly independent on $\mathbf{(}\mathbf{-}\mathbf{∞}\mathbf{,}\mathbf{∞}\mathbf{)}$.

(c). Verify that the function ${\mathbf{y}}_{\mathbf{3}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{|}\mathbf{t}{\mathbf{|}}^{\mathbf{3}}$is also a solution to 21on $\mathbf{(}\mathbf{-}\mathbf{∞}\mathbf{,}\mathbf{∞}\mathbf{)}$.

(d). Prove that there is no choice of constants ${\mathbf{c}}_{\mathbf{1}}\mathbf{,}{\mathbf{c}}_{\mathbf{2}}$such that ${\mathbf{y}}_{\mathbf{3}}\left(\mathbf{t}\right)\mathbf{=}{\mathbf{c}}_{\mathbf{1}}{\mathbf{y}}_{\mathbf{1}}\left(\mathbf{t}\right)\mathbf{+}{\mathbf{c}}_{\mathbf{2}}{\mathbf{y}}_{\mathbf{2}}\left(\mathbf{t}\right)$for all tin $\mathbf{(}\mathbf{-}\mathbf{∞}\mathbf{,}\mathbf{∞}\mathbf{)}$. [Hint: Argue that the contrary assumption leads to a contradiction.]

(e). From parts (c)and (d), we see that there is at least one solution to 21on $\mathbf{(}\mathbf{-}\mathbf{∞}\mathbf{,}\mathbf{∞}\mathbf{)}$that is not expressible as a linear combination of the solutions ${\mathbf{y}}_{\mathbf{1}}\left(\mathbf{t}\right)\mathbf{,}{\mathbf{y}}_{\mathbf{2}}\left(\mathbf{t}\right)$. Does this provide a counterexample to the theory in this section? Explain.

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

27. y₁(t) = costsint, y₂(t) = sin2t

Find a general solution to the given differential equation.

${\mathbf{x}}^{\mathbf{2}}\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{2}\mathbf{xy}\mathbf{\text{'}}\mathbf{-}\mathbf{2}\mathbf{y}\mathbf{=}\mathbf{6}{\mathbf{x}}^{\mathbf{-}\mathbf{2}}\mathbf{+}\mathbf{3}\mathbf{x}\mathbf{,}    \mathbf{x}\mathbf{>}\mathbf{0}$

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

28. y₁(t) = e^3t, y₂(t) = e^-4t

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

Find the solution to the initial value problem.

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{y}\mathbf{\text{'}}\mathbf{-}\mathbf{12}\mathbf{y}\mathbf{=}{\mathbf{e}}^t\mathbf{+}{\mathbf{e}}^{2t}\mathbf{-}\mathbf{1}\mathbf{;}\text{ â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰}\mathbf{y}\left(\mathbf{0}\right)\mathbf{=}\mathbf{1}\mathbf{,}\text{ â¶Ä‰â¶Ä‰â¶Ä‰}\mathbf{y}\mathbf{\text{'}}\left(\mathbf{0}\right)\mathbf{=}\mathbf{3}$