Question

Find the general solution of the given differential equation. $$ y^{\mathrm{iv}}+6 y^{\prime \prime \prime}+17 y^{\prime \prime}+22 y^{\prime}+14 y=0 $$

Short answer

Answer: The general solution is: \(y(x) = C_1e^{-x}+C_2e^{-2x}+e^{-x}[C_5\cdot cos(\sqrt{3}x)+C_6\cdot sin(\sqrt{3}x)]\), where \(C_1\), \(C_2\), \(C_5\), and \(C_6\) are constants.

Step-by-step solution

Write down the characteristic equation

The characteristic equation for the given differential equation is: $$ r^4 + 6r^3 + 17r^2 + 22r + 14 = 0 $$

Find the roots of the characteristic equation

To find the roots of the characteristic equation, you may use factoring, synthetic division, or an algorithm like the Durand-Kerner method. After finding the roots, it is found that the characteristic equation has roots \(r_1 = -1\), \(r_2 = -2\), \(r_3 = -1+\sqrt{3}i\), and \(r_4 = -1-\sqrt{3}i\).

Write the general solution using the roots

Using the roots of the characteristic equation, we can now write the general solution for the given differential equation: $$ y(x) = C_1e^{-x} + C_2e^{-2x} + C_3e^{(-1+\sqrt{3}i)x} + C_4e^{(-1-\sqrt{3}i)x} $$

Apply Euler's formula for the complex roots

We have two complex roots, \(-1+\sqrt{3}i\) and \(-1-\sqrt{3}i\). We can use Euler's formula to rewrite the terms with complex roots in the general solution: $$ e^{(-1+\sqrt{3}i)x} = e^{-x}(cos(\sqrt{3}x)+i\cdot sin(\sqrt{3}x)), \\ e^{(-1-\sqrt{3}i)x} = e^{-x}(cos(\sqrt{3}x)-i\cdot sin(\sqrt{3}x)) $$ Now, combine the two terms from Step 3 that have complex roots using these new expressions: $$ C_3e^{(-1+\sqrt{3}i)x} + C_4e^{(-1-\sqrt{3}i)x} = e^{-x}[(C_3+C_4)cos(\sqrt{3}x)+ (C_3-C_4)i\cdot sin(\sqrt{3}x)] $$ Let \(C_5 = C_3 + C_4\) and \(C_6 = C_3 - C_4\), so the combined term becomes: $$ e^{-x}[C_5\cdot cos(\sqrt{3}x)+ C_6\cdot sin(\sqrt{3}x)] $$

Write the final form of the general solution

Now we have the final form of the general solution: $$ y(x) = C_1e^{-x} + C_2e^{-2x} + e^{-x}[C_5\cdot cos(\sqrt{3}x)+ C_6\cdot sin(\sqrt{3}x)] $$ This is the general solution of the given fourth-order linear homogeneous differential equation.

Key concepts

Characteristic Equation

A characteristic equation is an essential tool when solving linear differential equations, especially those of higher order like the fourth-order differential equation in this problem. In general, a differential equation can be transformed into a characteristic equation by assuming a solution of the form \( y = e^{rx} \). Substituting this assumption into the differential equation allows us to express the equation as a polynomial in terms of \( r \). For the given equation: \[ y^{\mathrm{iv}}+6 y^{\prime \prime \prime}+17 y^{\prime \prime}+22 y^{\prime}+14 y=0 \]The corresponding characteristic equation is:\[ r^4 + 6r^3 + 17r^2 + 22r + 14 = 0 \]This equation's roots provide critical information needed for constructing the general solution of the differential equation. Each root corresponds to a fundamental solution that can be combined to form the overall solution.

Complex Roots

When solving the characteristic equation, we often encounter complex roots, especially in higher-order differential equations. Complex roots come in conjugate pairs, such as \( -1+\sqrt{3}i \) and \( -1-\sqrt{3}i \) as found in this exercise. These pairs are crucial because they indicate oscillatory behavior in the solutions due to their imaginary parts. To handle complex roots: - Identify the roots that form complex conjugate pairs.- Use them to form terms involving sine and cosine functions.The use of Euler's Formula can simplify the process by transforming exponential terms involving complex numbers into trigonometric functions, which represent the said oscillations. This is a key step in creating the general solution for the differential equation.

Euler's Formula

Euler's Formula is a bridge between exponential and trigonometric functions, allowing us to simplify expressions involving complex numbers. The formula states:\[ e^{ix} = \cos(x) + i\sin(x) \]For complex roots of a differential equation, Euler's formula enables us to express exponential terms in terms of sine and cosine.In this scenario, for a complex root \( -1+\sqrt{3}i \), applying Euler's formula yields:\[ e^{(-1+\sqrt{3}i)x} = e^{-x}[\cos(\sqrt{3}x) + i\sin(\sqrt{3}x)] \]This transformation helps us rewrite parts of the general solution in a more accessible form, highlighting the oscillatory component inherent in complex roots through the cosine and sine functions.

General Solution

The general solution of a differential equation is essentially a combination of solutions derived from its characteristic equation's roots. Each root brings a unique component to the solution. For the fourth-order differential equation we dealt with:- Real roots like \( r_1 = -1 \) and \( r_2 = -2 \) contribute terms of the form \( C_1e^{-x} \) and \( C_2e^{-2x} \). - Complex conjugate roots \( -1+\sqrt{3}i \) and \( -1-\sqrt{3}i \) are transformed using Euler's formula into terms involving sine and cosine.Finally, the full general solution is:\[ y(x) = C_1e^{-x} + C_2e^{-2x} + e^{-x}[C_5\cos(\sqrt{3}x) + C_6\sin(\sqrt{3}x)] \]This solution reflects the combination of exponential decay (from the real roots) and oscillatory behavior (from the complex roots) that is characteristic of many solutions to differential equations.