Math Problem Statement

Memorize this method

Form a DE whose general solution is given by y(x) = e^(-2x)(Acos2x + Bsin2x)

Solution

The images you've provided detail the method for forming a differential equation from a given general solution. I'll outline and summarize the key steps presented:

  1. Given general solution: y(x)=e2x(Acos2x+Bsin2x)y(x) = e^{-2x} (A\cos 2x + B\sin 2x)

  2. First derivative of y(x)y(x) with respect to xx: Differentiate the given function: y=2e2x(Acos2x+Bsin2x)+e2x(2Asin2x+2Bcos2x)y' = -2e^{-2x}(A\cos 2x + B\sin 2x) + e^{-2x}(-2A\sin 2x + 2B\cos 2x) Simplified form: y+2y=e2x(2Asin2x+2Bcos2x)y' + 2y = e^{-2x}(-2A\sin 2x + 2B\cos 2x)

  3. Second derivative of y(x)y(x): Differentiate again: y+2y=2e2x(2Asin2x+2Bcos2x)4yy'' + 2y' = -2e^{-2x}(-2A\sin 2x + 2B\cos 2x) - 4y Simplified expression for y+2yy'' + 2y': y+2y=2(y+2y)4yy'' + 2y' = -2(y' + 2y) - 4y

  4. Final form of the differential equation: After solving, we derive the second-order homogeneous linear differential equation: y+4y+8y=0y'' + 4y' + 8y = 0

This method uses differentiation and simplification to form a second-order differential equation that corresponds to the general solution provided.

Would you like further details or have any questions about this process? Here are five related questions you can explore:

  1. How do you derive a particular solution from a given general solution for a differential equation?
  2. Why is the homogeneous form of a differential equation important in this context?
  3. Can this method be applied to other forms of solutions involving exponential and trigonometric functions?
  4. How can you verify that the differential equation is correct by substitution?
  5. What are the common applications of this type of second-order differential equation?

Tip: Always simplify derivatives carefully, as small algebraic mistakes can propagate through subsequent steps, leading to incorrect final equations.

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Math Problem Analysis

Mathematical Concepts

Differential Equations
Exponential Functions
Trigonometric Functions

Formulas

y'(x) = -2e^(-2x)(Acos2x + Bsin2x) + e^(-2x)(-2Asin2x + 2Bcos2x)
y''(x) = Further differentiation of y'(x)

Theorems

Homogeneous Linear Differential Equation

Suitable Grade Level

Undergraduate Mathematics