Math Problem Statement
Solution
We are solving the given differential equation:
with initial conditions: and using as the independent variable.
Step 1: Rewrite the operator form
Let , so the equation becomes: Expanding, this represents a fourth-order differential equation.
Step 2: Solve the characteristic equation
The characteristic equation of is: Since are roots with multiplicity 2, the general solution to is: where are constants determined by initial conditions.
Step 3: Compute derivatives of
- First derivative:
- Second derivative:
- Third derivative:
Step 4: Apply initial conditions
- :
- :
- :
- :
Step 5: Solve for constants
From the equations:
- ,
- ,
- ,
Solve this system to find .
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Math Problem Analysis
Mathematical Concepts
Differential Equations
Characteristic Equation
Initial Value Problem
Formulas
Characteristic equation r^2 + 16 = 0
General solution: y(t) = (C1 + C2 t) cos(4t) + (C3 + C4 t) sin(4t)
Theorems
Superposition Principle
Complex Root Behavior in Differential Equations
Suitable Grade Level
College Level (Calculus II or Differential Equations)
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