Problem 304 Find the real-valued solution of... [FREE SOLUTION]

Chapter 14: Problem 304

Find the real-valued solution of $$ \left(D^{2}+2 D+4\right) y=0 . $$

Short Answer

Expert verified

The real-valued solution to the given second-order linear homogeneous differential equation $\left(D^{2}+2 D+4\right) y=0$ is $y(x) = e^{-x}(C_1\cos{\sqrt{3}x} + C_2\sin{\sqrt{3}x})$, where $C_1$ and $C_2$ are constants determined by initial/boundary conditions.

Step by step solution

Write down the characteristic equation

To find the characteristic equation, we replace the derivatives (D) with a variable m: $D^2$ becomes $m^2$, and $2D$ becomes $2m$. The given differential equation becomes \[ m^2 + 2m + 4 = 0. \]

Find the roots of the characteristic equation

To find the roots of the characteristic equation, we can use the quadratic formula: $m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where $a = 1$, $b = 2$, and $c = 4$. Plugging in the values, we get \[ m = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 1 \cdot 4}}{2 \cdot 1} = -1 \pm \frac{\sqrt{-12}}{2}. \]

Identify the nature of the roots

Since the discriminant, $b^2 - 4ac = -12$, is negative, we have complex conjugate roots: \[ m_1 = -1 + \frac{\sqrt{-12}}{2} = -1 + \frac{2\sqrt{3}i}{2} = - 1 + \sqrt{3}i \quad \text{and} \quad m_2 = -1 - \sqrt{3}i. \]

Write down the corresponding general solution

Since the roots are complex conjugates, the general solution will take the form \[ y(x) = e^{-x}(C_1\cos{\sqrt{3}x} + C_2\sin{\sqrt{3}x}), \] where $C_1$ and $C_2$ are constants determined by initial/boundary conditions (not provided in this exercise).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Complex roots

Complex roots appear when solving quadratic equations with a negative discriminant. In the context of differential equations, they're crucial for understanding the behavior of solutions. A complex conjugate pair has the form

$ m_1 = a + bi $
$ m_2 = a - bi $

where $a$ is the real part and $bi$ is the imaginary part. In our problem, after calculating the discriminant $b^2 - 4ac = -12$, we identified complex roots:

$ m_1 = -1 + \sqrt{3}i $
$ m_2 = -1 - \sqrt{3}i $

These roots signify that the solution involves exponential decay (due to $-1$) and oscillatory components (due to $ \sqrt{3}i $).
This is typical for stable systems that oscillate around a particular value.

Quadratic formula

The quadratic formula is a valuable tool for solving second-degree polynomial equations. Given a quadratic equation expressed as $ax^2 + bx + c = 0$, the roots $m$ can be found using:\[m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]By substituting the coefficients $a = 1$, $b = 2$, and $c = 4$ from our specific characteristic equation, we derived:

$ m = \frac{-2 \pm \sqrt{2^2 - 4 \times 1 \times 4}}{2 \times 1} = -1 \pm \frac{\sqrt{-12}}{2} $

Using the quadratic formula allows us to systematically uncover the equation's roots. In our case, these roots are complex, indicating oscillatory behavior in the solution. The negative discriminant is the telltale indicator of complex roots.

Homogeneous linear differential equation

A homogeneous linear differential equation is one in which the sum of its terms equals zero. In such equations, all terms rely on the function or its derivatives. The equation:$ \left(D^{2} + 2 D + 4\right) y = 0 $is considered homogeneous because it equates to zero. This specific form implies that any combination of solutions still satisfies the equation.
The coefficients in front of the derivatives (like $D^2$ and $2D$) lead to a characteristic equation. Solving this helps find particular solutions, represented by exponential functions tied to the roots we determined.

Simple and structured, these problems follow established patterns, often leading to generalized solutions like ours.

General solution

The general solution of a differential equation encompasses all possible solutions using arbitrary constants. For our equation with complex roots, the general solution takes the form:\[y(x) = e^{-x}(C_1\cos{\sqrt{3}x} + C_2\sin{\sqrt{3}x})\]Here:

$ e^{-x} $ reflects the exponential decay influenced by the real part of the roots.
$ \cos{\sqrt{3}x} $ and $ \sin{\sqrt{3}x} $ represent oscillatory behavior from the imaginary parts of the roots.

Constants $C_1$ and $C_2$ are vital. These are usually found using initial or boundary conditions. Until such conditions are provided, the solution remains a broad representation, covering every possible scenario.

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91影视

Find the real-valued solution of $$ \left(D^{2}+2 D+4\right) y=0 . $$

Short Answer

Step by step solution

Write down the characteristic equation

Find the roots of the characteristic equation

Identify the nature of the roots

Write down the corresponding general solution

Key Concepts

Complex roots

Quadratic formula

Homogeneous linear differential equation

General solution

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