Math Problem Statement

Solve the expression involving differential operators and simplify the given function with respect to trigonometric input.

Solution

The uploaded image appears to contain a worked mathematical problem, likely involving differential operators. Let me provide a detailed breakdown of the visible solution.


Key Observations:

The equation involves the following:

  1. A differential operator DD, where DnD^n implies the nn-th derivative with respect to xx.
  2. An exponential term e2xe^{-2x}.
  3. A trigonometric term cos3x\cos 3x.
  4. The operator is applied to simplify an expression and solve for a particular solution.

Steps from the Image:

  1. Expression Simplification: The term initially shown is 1D24D+41D21cos3x\frac{1}{D^2 - 4D + 4} \cdot \frac{1}{D^2 - 1} \cos 3x This is combined into: =1(D2)21D21cos3x= \frac{1}{(D - 2)^2} \cdot \frac{1}{D^2 - 1} \cos 3x

  2. Applying the Trigonometric Input: The equation indicates that the operators act on cos3x\cos 3x. To proceed, recall: D2cos3x=9cos3xD^2 \cos 3x = -9 \cos 3x Substituting into D21D^2 - 1, we get: D21(9)1=10.D^2 - 1 \to (-9) - 1 = -10.

  3. Combining Results: The solution incorporates the exponential factor e2xe^{-2x}. The fraction simplifies with the remaining operator calculations: =110e2xcos3x= \frac{1}{-10} e^{-2x} \cos 3x

  4. Final Answer: The simplified result is: =120e2xcos3x= -\frac{1}{20} e^{-2x} \cos 3x


Would you like me to:

  1. Explain differential operators in detail?
  2. Derive the trigonometric operator result D2cos3x=9cos3xD^2 \cos 3x = -9 \cos 3x?
  3. Explore why (D2)2(D - 2)^2 arises in this context?
  4. Provide more examples of operator-based solutions?
  5. Solve a related differential equation?

Tip:

When dealing with DnD^n and trigonometric functions, always compute derivatives explicitly to verify operator simplifications.

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Math Problem Analysis

Mathematical Concepts

Differential Operators
Exponential Functions
Trigonometric Functions

Formulas

D^2 cos(kx) = -k^2 cos(kx)
Fraction simplification of differential operators

Theorems

Properties of Differential Operators
Simplification of Linear Operators

Suitable Grade Level

Undergraduate (Mathematics or Engineering)