Q Review Problems-7E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

Find a differential operator that annihilates the given function.

(a)  x² - 2x + 5

(b)  e^3x + x - 1

(c)  x sin2x

(d)  x²e^-2x cos3x

(e)  x² - 2x + xe^-x + sin2x - cos3x

Step-by-Step Solution

Answer

The solution for the differential operator that annihilates the given function:

1.  D²⁺¹ = D³
2.  D² (D - 3)
3.  (D² + 2²) = (D² + 4)²
4.  [(D + 2)² + 3² ]³ = (D² + 4D + 4 + 9)³ = (D² + 4D + 13)³
5.  D³(D + 1)² (D² + 2²) (D² + 3²) = D³(D + 1)² (D² + 4) (D² + 9)

Step 1: Determine the differential operator that annihilates for the given function.

Consider the given function:

 $f\left(x\right)=x^2-2x+5$

Since f(x)  is a polynomial function of second order then it is annihilated by:

 $D^{2+1}=D^3$

Step 2: Determine the differential operator that annihilates for the given function.

Consider the given function:

 $f\left(x\right)=e^{3x}+x-1.$

Let  $f_1\left(x\right)=e^{3x}$ and $f_2\left(x\right)=x-1$. Observe that D - 3 annihilates f₁(x) and  f₂(x)  is annihilated by D². 

Hence, the composite operator $D^2(D-3)$ annihilates both f₁(x) and  f₂(x)  so it annihilates 

f₁(x) + f₂(x).

Step 3: Determine the differential operator that annihilates for the given function.

Consider the given function:

 $f\left(x\right)=x\sin 2x$

This function is annihilated by:

 ${\left(D^2+2^2\right)}^2={\left(D^2+4\right)}^2.$

Step 4: Determine the differential operator that annihilates for the given function.

Consider the given function:

 $f\left(x\right)=x^2{e}^{-2x}\cos 3x$

This function is annihilated by:

[(D + 2)² + 3² ]³ = (D² + 4D + 4 + 9)³ = (D² + 4D + 13)³

Step 5: Determine the differential operator that annihilates for the given function .

Consider the given function:

 $f\left(x\right)=x^2-2x+x{e}^{-x}+\sin 2x-\cos 3x$

Let  $f_1\left(x\right)=x^2-2x,f_2\left(x\right)=x{e}^{-x},f_3\left(x\right)=\sin 2x$ and $f_4\left(x\right)=\cos 3x$.

Observe that D³  annihilates   f₁(x) and  f₂(x)    is annihilated by (D + 1)².

Furthermore, we see that D² + 2²  annihilates   f₃(x) and  f₄(x)   is annihilated by D² + 3² . Hence, the composite operator D³(D + 1)² (D² + 2²) (D² + 3²) = D³(D + 1)² (D² + 4) (D² + 9)

annihilates  f₁(x), f₂(x), f₃(x) and  f₄(x) so it annihilates f₁(x) + f₂(x) + f₃(x) - f₄(x)