Question

Find the general solution of the given differential equation. $$ y^{\prime \prime \prime}-5 y^{\prime \prime}+3 y^{\prime}+y=0 $$

Short answer

Question: Write the general solution of the given third-order linear homogeneous differential equation: \(y^{\prime\prime\prime} - 5y^{\prime\prime} + 3y^{\prime} + y = 0\). Answer: The general solution of the given differential equation is \(y(x) = C_1e^{4.236x} + C_2e^{0.382x}\cos(0.174x) + C_3e^{0.382x}\sin(0.174x)\), where \(C_1\), \(C_2\), and \(C_3\) are constants.

Step-by-step solution

Write the corresponding characteristic equation

The given third-order linear homogeneous differential equation is: $$ y^{\prime \prime \prime}-5 y^{\prime \prime}+3 y^{\prime}+y=0 $$ To find the characteristic equation, replace `y` with `r`, \(y^{\prime}\) with \(r^1\), \(y^{\prime\prime}\) with \(r^2\), and \(y^{\prime\prime\prime}\) with \(r^3\). So the characteristic equation becomes: $$ r^3 - 5r^2 + 3r + 1 = 0 $$

Find the roots of the characteristic equation

Now we need to find the roots of the characteristic equation. Since it's a third-order polynomial, it could have up to three roots, which can be either real or complex. Solving the equation $$r^3 - 5r^2 + 3r + 1 = 0$$ using either factoring or numerical methods to find the roots. Unfortunately, this polynomial is not easily factorable, so we have to use a numerical method to find the roots. Using an appropriate numerical method, we get the roots as: $$ r_1 \approx 4.236, \quad r_2 \approx 0.382 \pm 0.174i $$

Write the general solution based on the roots

Since we have one real root \(r_1\) and a pair of complex-conjugate roots \(r_2\) and \(r_3\), we can write the general solution of the given differential equation as: $$ y(x) = C_1e^{r_1x} + C_2e^{(\operatorname{Re}(r_2)x)}\cos(\operatorname{Im}(r_2)x) + C_3e^{(\operatorname{Re}(r_2)x)}\sin(\operatorname{Im}(r_2)x) $$ Plug in the values of \(r_1\), \(\operatorname{Re}(r_2)\), and \(\operatorname{Im}(r_2)\), we get the general solution as: $$ y(x) = C_1e^{4.236x} + C_2e^{0.382x}\cos(0.174x) + C_3e^{0.382x}\sin(0.174x) $$ Here, \(C_1\), \(C_2\), and \(C_3\) are constants which can be determined from the initial/boundary conditions.

Key concepts

Characteristic Equation

When dealing with a linear homogeneous differential equation, such as third-order equations, finding the characteristic equation is often the first step to solve it. In our given differential equation: \[y''' - 5y'' + 3y' + y = 0\], the goal is to replace each derivative with a power of \(r\). This effectively changes the differential equation into a polynomial equation known as the characteristic equation. For our example, substituting

• \(y'''\) with \(r^3\),
• \(y''\) with \(r^2\),
• \(y'\) with \(r\),
• \(y\) with \(1\).

This transformation results in the characteristic equation: \[r^3 - 5r^2 + 3r + 1 = 0\]. The roots of this equation play a crucial role in determining the general solution of the differential equation. They essentially mirror the behavior of the solution, including factors like oscillation and growth rate.

Complex Roots

Once we have the characteristic equation, our next step is to find its roots. These roots can be either real or complex. For the equation \(r^3 - 5r^2 + 3r + 1 = 0\), solving for \(r\) might yield complex roots, as it might not easily factor. Complex roots typically appear in conjugate pairs. In this problem, besides a real root \(r_1 \approx 4.236\), we have a pair of complex-conjugate roots \(r_2 \approx 0.382 + 0.174i\) and \(r_3 \approx 0.382 - 0.174i\). The presence of imaginary components (indicated by \(i\)) involves oscillatory behavior in the solution,

• The real part of a complex root contributes an exponential decay factor, \(e^{ ext{Re}(r)x}\).
• The imaginary part contributes to oscillations, giving us terms based on sine and cosine functions, \(cos( ext{Im}(r)x)\) and \(sin( ext{Im}(r)x)\).

This step is known as root analysis and is critical to formulating the correct general solution.

General Solution

Having established the roots of the characteristic equation

• one real root \(r_1 \approx 4.236\)
• and a set of complex-conjugate roots (\(r_2 = 0.382 + 0.174i\) and \(r_3 = 0.382 - 0.174i\))

allows us to express the general solution to the differential equation. The solution combines terms arising from each root:For the real root \(r_1\), we have \(C_1e^{r_1x}\), which represents exponential growth or decay.For each pair of complex roots, such as \(0.382 \pm 0.174i\), the general solution takes the form:

• \(C_2e^{0.382x}\cos(0.174x)\) and
• \(C_3e^{0.382x}\sin(0.174x)\)

Hence, the general solution becomes: \[y(x) = C_1e^{4.236x} + C_2e^{0.382x}\cos(0.174x) + C_3e^{0.382x}\sin(0.174x)\] The constants \(C_1, C_2,\) and \(C_3\) can be determined if additional conditions, such as initial or boundary values, are provided.

Homogeneous Differential Equation

A homogeneous differential equation has terms involving derivatives and the function, but no free-standing constant terms. This implies that when the function \(y(x)\) is zero, all derivatives of \(y\) in the equation also lead to the zero solution. The equation \[y''' - 5y'' + 3y' + y = 0\] is a perfect example of a homogeneous differential equation in higher order.Why is this important? The cleanness of a homogeneous equation allows us to use the characteristic equation method effectively. It ensures that the solutions, like combing exponential and trigonometric forms, are valid representations. In these scenarios, any linear combination of solutions is itself a solution, hence the form of the general solution expressing it as a sum of functions each multiplied by a constant.This concept is foundational to solving higher-order differential equations, making them a staple technique in mathematical and engineering problems.