Q Review Problems-2E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

Determine whether the given functions are linearly dependent or linearly independent on the interval $\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{\infty }\mathbf{)}$ .

(a)  $\left\{e^{2x},x^2{e}^{2x},e^{-x}\right\}$

(b)  $\left\{e^{x}sin2x,x{e}^{x}sin2x,e^x,x{e}^x\right\}$

(c)   $\left\{2e^{2x}-e^x,e^{2x}+1,e^{2x}-3,e^x+1\right\}$

Step-by-Step Solution

Answer

1. Given set of functions are Linearly Independent (LI).
2. Given set of functions are Linearly Independent (LI).
3. Given set of functions are Linearly Independent (LI).

Step 1: Determine the given functions are linearly dependent or linearly independent

$<e^{2x},x^2{e}^{2x},e^{-x}>$

Assuming scalars  $c_1,c_2,c_3$ such that

 $c_1{e}^{2x}+c_2{x}^2{e}^{2x}+c_3{e}^{-x}=0$

At 

$$\begin{gathered} x=0 \\ {c}_4=0=c_2 \\ {c}_1+c_3=0 \end{gathered}$$

At  $x\to \infty$

 $$\begin{gathered} c_1(\infty )+c_2(\infty )=0 \\ {c}_1=c_2=0=c_3 \end{gathered}$$

Hence

Given set of functions are Linearly Independent (LI).

Step 2: Determine the given functions are linearly dependent or linearly independent

$<e^x\sin 2x,x{e}^x\sin 2x,e^x,x{e}^x>$

Assuming scalars  $c_1,c_2,c_3,c_4$ such that

 $c_1{e}^x\sin 2x+c_{2}x{e}^x\sin 2x+c_3{e}^x+c_{4}x{e}^x=0$

At  $x\to 0$

 $c_4=0=c_2$

At  x = 0

$$\begin{gathered} c_3=0 \\ {c}_1=c_2=c_3=c_4=0 \end{gathered}$$

 

Hence

Given set of functions are Linearly Independent (LI).

Step 3: Determine the given functions are linearly dependent or linearly independent

$<2e^{2x}-e^x,e^{2x}+1,e^{2x}-3,e^x+1>$

Assuming scalars  $c_1,c_2,c_3,c_4$ such that

$c_1\left(2e^{2x}-e^x\right)+c_2\left(e^{2x}+1\right)+c_3\left(e^{2x}-3\right)+c_4\left(e^x+1\right)=0$

At  $x\to -\infty$

 $c_2-2c_3+c_4=0$

At  x = 0

$c_1+2c_2-2c_3+2c_4=0$

 

At  x = 1

$e^2\left(2c_1+c_2+c_3\right)=c_1-c_1$

At x = -1

$\left.2c_1+c_2+c_3+c_{\left(c_1\right.}-c_1\right)=0$

From equation third and fourth we get

 

$$\begin{gathered} c_2=-c_1 \\ {c}_4=c_1 \\ {c}_3=-c_1 \end{gathered}$$

From equation second, we can get

 $$\begin{gathered} c_1=0 \\ {c}_1=c_2=c_3=c_4=0 \end{gathered}$$

Hence

Given set of functions are Linearly Independent (LI).