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Magazine Chapter 31: Problem 22
Solve the given differential equations. $$D^{2} y+4 D y+6 y=0$$
Short Answer
Expert verified
The general solution is \( y(x) = e^{-2x}(C_1 \cos(\sqrt{2} x) + C_2 \sin(\sqrt{2} x)) \).
Step by step solution
01
Identify the Type of Differential Equation
The given differential equation is a second-order linear homogeneous differential equation. It is characteristic of the form \( aD^2 y + bD y + cy = 0 \), where the coefficients \( a = 1 \), \( b = 4 \), and \( c = 6 \).
02
Write the Characteristic Equation
To solve the differential equation, we find its characteristic equation by replacing \( D^n \) with \( m^n \). This gives us \( m^2 + 4m + 6 = 0 \).
03
Solve the Characteristic Equation
Next, we solve the quadratic equation \( m^2 + 4m + 6 = 0 \) using the quadratic formula: \( m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = 4 \), and \( c = 6 \).
04
Calculate the Discriminant
Calculate the discriminant \( b^2 - 4ac \) for the quadratic equation: \( 4^2 - 4 \times 1 \times 6 = 16 - 24 = -8 \). Since the discriminant is negative, the roots will be complex conjugates.
05
Determine the Roots
Using the quadratic formula and the calculated discriminant, the roots are \( m = \frac{-4 \pm \sqrt{-8}}{2} \). Simplifying gives \( m = -2 \pm i \sqrt{2} \).
06
Write the General Solution
The general solution for a differential equation with complex conjugate roots \( m = \alpha \pm i \beta \) is \( y = e^{\alpha x}(C_1 \cos(\beta x) + C_2 \sin(\beta x)) \). Substitute \( \alpha = -2 \) and \( \beta = \sqrt{2} \) to get \( y(x) = e^{-2x}(C_1 \cos(\sqrt{2} x) + C_2 \sin(\sqrt{2} x)) \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Characteristic Equation
In solving second-order linear homogeneous differential equations, a crucial step is forming the characteristic equation. This equation helps us find the roots that determine the solution's behavior. To form the characteristic equation, we substitute each derivative operator \( D^n \) in the differential equation with \( m^n \), where \( m \) represents the roots of the equation. For the given example, the original equation \( D^2y + 4Dy + 6y = 0 \) becomes the characteristic equation \( m^2 + 4m + 6 = 0 \).\
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- Purpose: Helps determine the nature of the solution (real or complex roots). \
- Type: Quadratic equation, giving insights into the roots' properties. \
- Steps: Transforms the differential equation by replacing each \( D \) with \( m \). \
Complex Roots
In the context of differential equations, encountering complex roots is common, especially when dealing with oscillatory solutions or when the discriminant of the characteristic equation is negative. For a quadratic equation like \( m^2 + 4m + 6 = 0 \), the discriminant is calculated using \( b^2 - 4ac \). If this value is negative, it indicates that the roots will be complex conjugates. For our problem, the discriminant \( 16 - 24 = -8 \) confirms this scenario.\
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- Result: Complex roots arise when the discriminant \( b^2 - 4ac < 0 \). \
- Complex roots form: \( m = \alpha \pm i \beta \), with \( i \) representing the imaginary unit. \
- Example roots: \( m = -2 \pm i\sqrt{2} \) in the given problem. \
Homogeneous Equations
A homogeneous differential equation is characterized by the absence of standalone terms; all terms involve the function or its derivatives. This type of equation is key in determining the general behavior of systems described by differential equations. The equation \( D^2y + 4Dy + 6y = 0 \) is an example, showing no terms solely in \( x \) or constants.\
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- Core Feature: Every term includes the dependent variable or its derivatives. \
- Solution Form: The solution usually involves exponential functions that are influenced by the equation's roots. \
- Impact of Characteristics: Roots of the characteristic equation directly determine the form of the solution. \
