Question

Find the function $\mathbf{f}\left(t\right)$of the form $\mathbf{f}\left(t\right)\mathbf{=}\mathbf{acos}\left(2t\right)\mathbf{+}\mathbf{bsin}\left(2t\right)$such that${\mathbf{f}}^{\mathbf{\text{'}}\mathbf{\text{'}}}\left(t\right)\mathbf{+}\mathbf{2}{\mathbf{f}}^{\mathbf{\text{'}}}\left(t\right)\mathbf{+}\mathbf{3}\mathbf{f}\left(t\right)\mathbf{=}\mathbf{17}\mathbf{cos}\left(2t\right)$. (This is the kind of differential equation you might have to solve when dealing with forced damped oscillators, in physics or engineering.)

Short answer

The function $\mathrm{f}\left(\mathrm{t}\right)$of the form $\mathrm{f}\left(\mathrm{t}\right)=\mathrm{acos}\left(2\mathrm{t}\right)+\mathrm{bsin}\left(2\mathrm{t}\right)$such that $\mathrm{f}\text{'}\text{'}\left(\mathrm{t}\right)+2{\mathrm{f}}^{\text{'}}\left(\mathrm{t}\right)+3\mathrm{f}\left(\mathrm{t}\right)=17\cos \left(2\mathrm{t}\right)$is$\mathrm{f}\left(\mathrm{t}\right)=-\cos \left(2\mathrm{t}\right)+4\sin \left(2\mathrm{t}\right)$

Step-by-step solution

Consider the standard equation.

The polynomial of degree 2 is $\mathrm{f}\left(\mathrm{t}\right)=\mathrm{a}+\mathrm{bt}+{\mathrm{ct}}^2$

$\mathrm{f}\left(\mathrm{t}\right)=\mathrm{acos}\left(2\mathrm{t}\right)+\mathrm{bsin}\left(2\mathrm{t}\right)$

Consider the derivative of the polynomial $\mathrm{f}\left(\mathrm{t}\right)=\mathrm{acos}\left(2\mathrm{t}\right)+\mathrm{bsin}\left(2\mathrm{t}\right)$

${\mathrm{f}}^{\text{'}}\left(\mathrm{t}\right)=-2\mathrm{asin}\left(2\mathrm{t}\right)+2\mathrm{bcos}\left(2\mathrm{t}\right)$

Consider the derivative of the polynomial${\mathrm{f}}^{\text{'}}\left(\mathrm{t}\right)=-2\mathrm{asin}\left(2\mathrm{t}\right)+2\mathrm{bcos}\left(2\mathrm{t}\right)$

${\mathrm{f}}^{\text{'}\text{'}}\left(\mathrm{t}\right)=-4\mathrm{acos}\left(2\mathrm{t}\right)-4\mathrm{bsin}\left(2\mathrm{t}\right)$

Consider the given condition

Consider the condition and substitute the above equations.

$\begin{array}{rcl}{\mathrm{f}}^{\text{'}\text{'}}\left(\mathrm{t}\right)+2{\mathrm{f}}^{\text{'}}\left(\mathrm{t}\right)+3\mathrm{f}\left(\mathrm{t}\right)& =& 17\cos \left(2\mathrm{t}\right)\\ \left[-4\mathrm{acos}\left(2\mathrm{t}\right)-4\mathrm{bsin}\left(2\mathrm{t}\right)\right]+2\left[-2\mathrm{asin}\left(2\mathrm{t}\right)+2\mathrm{bcos}\left(2\mathrm{t}\right)\right]+3\left[\mathrm{acos}\left(2\mathrm{t}\right)+\mathrm{bsin}\left(2\mathrm{t}\right)\right]& =& 17\cos \left(2\mathrm{t}\right)\\ -\mathrm{acos}\left(2\mathrm{t}\right)-\mathrm{bsin}\left(2\mathrm{t}\right)-4\mathrm{asin}\left(2\mathrm{t}\right)+4\mathrm{bcos}\left(2\mathrm{t}\right)& =& 17\cos \left(2\mathrm{t}\right)\\ \left(4\mathrm{b}-\mathrm{a}\right)\cos \left(2\mathrm{t}\right)+\left(-4\mathrm{a}-\mathrm{b}\right)\sin \left(2\mathrm{t}\right)& =& 17\cos \left(2\mathrm{t}\right)\end{array}$

Compare the LHS and RHS of the above equations.

The values are:

$$\begin{gathered} -\mathrm{a}+4\mathrm{b}=17 \\ -4\mathrm{a}-\mathrm{b}=0 \\ \therefore \mathrm{a}=-1,\mathrm{b}=4 \end{gathered}$$

Substitute the values in the equation.

$$\begin{gathered} \mathrm{f}\left(\mathrm{t}\right)=\mathrm{acos}\left(2\mathrm{t}\right)+\mathrm{bsin}\left(2\mathrm{t}\right) \\ \mathrm{f}\left(\mathrm{t}\right)=-\cos \left(2\mathrm{t}\right)+4\sin \left(2\mathrm{t}\right) \end{gathered}$$

Final answer

$\mathrm{f}\left(\mathrm{t}\right)=-\cos \left(2\mathrm{t}\right)+4\sin \left(2\mathrm{t}\right)$is the function$\text{f}\left(\mathrm{t}\right)$ of the form$\mathrm{f}\left(\mathrm{t}\right)=\mathrm{acos}\left(2\mathrm{t}\right)+\mathrm{bsin}\left(2\mathrm{t}\right)$ such that${\mathrm{f}}^{\text{'}\text{'}}\left(\mathrm{t}\right)+2{\mathrm{f}}^{\text{'}}\left(\mathrm{t}\right)+3\mathrm{f}\left(\mathrm{t}\right)=17\cos \left(2\mathrm{t}\right)$.