Q11E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

In Problems 3 – 18, use the elimination method to find a general solution for the given linear system, where differentiation is with respect to t.

$$\begin{gathered} \left({\mathbf{D}}^{\mathbf{2}}\mathbf{-}\mathbf{1}\right)\left[\mathbf{u}\right]\mathbf{+}\mathbf{5}\mathbf{v}\mathbf{=}{\mathbf{e}}^{\mathbf{t}}\mathbf{,} \\ \mathbf{2}\mathbf{u}\mathbf{+}\left({\mathbf{D}}^{\mathbf{2}}\mathbf{+}\mathbf{2}\right)\left[\mathbf{v}\right]\mathbf{=}\mathbf{0} \end{gathered}$$

Step-by-Step Solution

Answer

The solutions for the given linear system are

 $u\left(t\right)=c_1{e}^{-\sqrt{3}t}+c_2{e}^{\sqrt{3}t}+c_3\cos \left(2t\right)+c_4\sin \left(2t\right)-\frac{3}{10}{e}^t$ and $v\left(t\right)=-\frac{2}{5}{c}_1{e}^{-\sqrt{3}t}-\frac{2}{5}{c}_2{e}^{\sqrt{3}t}+c_3\cos \left(2t\right)+c_4\sin \left(2t\right)+\frac{e^t}{5}$

.

Step 1: General form

Elimination Procedure for 2 × 2 Systems

 

To find a general solution for the system

 $$\begin{gathered} {\mathrm{L}}_1\left[\mathrm{x}\right]+{\mathrm{L}}_2\left[\mathrm{y}\right]={\mathrm{f}}_1, \\ {\mathrm{L}}_3\left[\mathrm{x}\right]+{\mathrm{L}}_4\left[\mathrm{y}\right]={\mathrm{f}}_2, \end{gathered}$$

1.  Where ${\mathbf{L}}_{\mathbf{1}}\mathbf{,}{\mathbf{L}}_{\mathbf{2}}\mathbf{,}{\mathbf{L}}_{\mathbf{3}}\mathbf{,}$ and ${\mathbf{L}}_{\mathbf{4}}$ are polynomials in $D=\frac{d}{dt}$

 

1. Make sure that the system is written in operator form.
2. Eliminate one of the variables, say, y, and solve the resulting equation for. If the system is degenerating, stop! A separate analysis is required to determine whether or not there are solutions.
3. (Shortcut) If possible, use the system to derive an equation that involves  but not its derivatives. [Otherwise, go to step (d).] Substitute the found expression for  into this equation to get a formula for. The expressions for , and  give the desired general solution.
4. Eliminate x from the system and solve for. [Solving for  gives more constants----in fact, twice as many as needed.]
5. Remove the extra constants by substituting the expressions for x(t) and  into one or both of the equations in the system. Write the expressions for  and  in terms of the remaining constants.

Step 2: Evaluate the given equation

Given that,

$\left(D^2-1\right)\left[u\right]+5v=e^t ......\mathbf{\left(}\mathbf{1}\mathbf{\right)}$

 $2u+\left(D^2+2\right)\left[v\right]=0 ......\mathbf{\left(}\mathbf{2}\mathbf{\right)}$

Multiply $\left(D^2+2\right)$ on equation (1).

$$\begin{gathered} \left(D^2+2\right)\left(D^2-1\right)\left[u\right]+5\left(D^2+2\right)v=\left(D^2+2\right)e^t \\ \left(D^2+2\right)\left(D^2-1\right)\left[u\right]+5\left(D^2+2\right)v=e^t+2e^t \\ \left(D^2+2\right)\left(D^2-1\right)\left[u\right]+5\left(D^2+2\right)v=3e^t ......\mathbf{\left(}\mathbf{3}\mathbf{\right)} \end{gathered}$$ 

 And multiply 5 on equation (2).

 $$\begin{gathered} 2\left(5\right)u+5\left(D^2+2\right)\left[v\right]=0 \\ 10u+5\left(D^2+2\right)\left[v\right]=0 ......\mathbf{\left(}\mathbf{4}\mathbf{\right)} \end{gathered}$$

 Then subtract equation (3) and (4) together one gets,

 $$\begin{gathered} \left(D^2+2\right)\left(D^2-1\right)\left[u\right]-10u=3e^t \\ \left(D^4-D^2+2D^2-2-10\right)\left[u\right]=3e^t \\ \left(D^4+D^2-12\right)\left[u\right]=3e^t \\ \left(D^2+4\right)\left(D^2-3\right)\left[u\right]=3e^t \end{gathered}$$

$\left(D^2+4\right)\left(D^2-3\right)\left[u\right]=3e^t ......\mathbf{\left(}\mathbf{5}\mathbf{\right)}$

 Since the auxiliary equation to the corresponding homogeneous equation is:

$\left(r^2+4\right)\left(r^2-3\right)=0$ 

. The roots are $r=\pm 2i$ and $r=\pm \sqrt{3}$.

 

Then, the homogeneous solution of u is 

$u_h\left(t\right)=c_1{e}^{-\sqrt{3}t}+c_2{e}^{\sqrt{3}t}+c_3\cos \left(2t\right)+c_4\sin \left(2t\right) ......\mathbf{\left(}\mathbf{6}\mathbf{\right)}$

 

Let us take the undetermined coefficients and assume that 

$u_p\left(t\right)=a{e}^t ......\mathbf{\left(}\mathbf{7}\mathbf{\right)}$

 

Now derivate the equation (7)

$$\begin{gathered} D^2\left[u_p\left(t\right)\right]=a{e}^t \\ {D}^4\left[u_p\left(t\right)\right]=a{e}^t \end{gathered}$$ 

Step 3: Substitution method

Substitute the derivation in equation (5).

$$\begin{gathered} \left(D^4+D^2-12\right)\left[u\right]=3e^t \\ \left(D^4+D^2-12\right)\left[a{e}^t\right]=3e^t \\ a{e}^t+a{e}^t-12a{e}^t=3e^t \\ -10a{e}^t=3e^t \end{gathered}$$ 

 Now, equalize the like terms.

$$\begin{gathered} -10a=3 \\ a=-\frac{3}{10} \end{gathered}$$ 

 So, $u_p\left(t\right)=-\frac{3}{10}{e}^t ......\mathbf{\left(}\mathbf{8}\mathbf{\right)}$

 

Use equations (6) and (8) to get,

 $$\begin{gathered} u\left(t\right)=u_h\left(t\right)+u_p\left(t\right) \\ u\left(t\right)=c_1{e}^{-\sqrt{3}t}+c_2{e}^{\sqrt{3}t}+c_3\cos \left(2t\right)+c_4\sin \left(2t\right)-\frac{3}{10}{e}^t ......\mathbf{\left(}\mathbf{9}\mathbf{\right)} \end{gathered}$$

 Now, take equation (1).

$$\begin{gathered} \left(D^2-1\right)\left[u\right]+5v=e^t \\ 5v=e^t-\left(D^2-1\right)\left[u\right] \\ v=\frac{e^t-\left(D^2-1\right)\left[u\right]}{5} \end{gathered}$$ 

 Now derivate the value of u to find the value of v.

 $$\begin{gathered} u\text{'}\left(t\right)=-\sqrt{3}{c}_1{e}^{-\sqrt{3}t}+\sqrt{3}{c}_2{e}^{\sqrt{3}t}-2c_3\sin \left(2t\right)+2c_4\cos \left(2t\right)-\frac{3}{10}{e}^t \\ u\text{'}\text{'}\left(t\right)=3c_1{e}^{-\sqrt{3}t}+3c_2{e}^{\sqrt{3}t}-4c_3\cos \left(2t\right)-4c_4\sin \left(2t\right)-\frac{3}{10}{e}^t \end{gathered}$$

Then,

$$\begin{gathered} v=\frac{e^t}{5}-\frac{\left(D^2-1\right)\left[u\right]}{5} \\ =\frac{e^t}{5}-\frac{\left(D^2-1\right)\left[c_1{e}^{-\sqrt{3}t}+c_2{e}^{\sqrt{3}t}+c_3\cos \left(2t\right)+c_4\sin \left(2t\right)-\frac{3}{10}{e}^t\right]}{5} \\ =\frac{e^t}{5}-\frac{3c_1{e}^{-\sqrt{3}t}+3c_2{e}^{\sqrt{3}t}-4c_3\cos \left(2t\right)-4c_4\sin \left(2t\right)-\frac{3}{10}{e}^t-\left(c_1{e}^{-\sqrt{3}t}+c_2{e}^{\sqrt{3}t}+c_3\cos \left(2t\right)+c_4\sin \left(2t\right)-\frac{3}{10}{e}^t\right)}{5} \\ =\frac{e^t}{5}-\frac{2c_1{e}^{-\sqrt{3}t}+2c_2{e}^{\sqrt{3}t}-5c_3\cos \left(2t\right)-5c_4\sin \left(2t\right)}{5} \\ =-\frac{2}{5}{c}_1{e}^{-\sqrt{3}t}-\frac{2}{5}{c}_2{e}^{\sqrt{3}t}+c_3\cos \left(2t\right)+c_4\sin \left(2t\right)+\frac{e^t}{5} \end{gathered}$$

So, the solution is founded.