Question

Find the solution of the given initial value problem and plot its graph. How does the solution behave as $t\to \mathrm{\infty }?$ $$\begin{array}{l}{y}^{\mathrm{v}}+6y^{\prime \prime \prime }+17y^{\prime \prime }+22y^{\prime }+14y=0;y(0)=1,y^{\prime }(0)=-2,y^{\prime \prime }(0)=0\\ y^{\prime \prime \prime }(0)=3\end{array}$$

Short answer

Answer: The general solution of a fourth-order linear homogeneous differential equation is given by $y(t)=C_1{e}^{r_{1}t}+C_2{e}^{r_{2}t}+C_3{e}^{r_{3}t}+C_4{e}^{r_{4}t}$, where $C_1,C_2,C_3$, and $C_4$ are constants, and $r_1,r_2,r_3,r_4$ are the roots of the corresponding characteristic equation. To find the particular solution, you can apply the initial conditions to the general solution and solve for the constants $C_1,C_2,C_3$, and $C_4$. After finding the specific values for these constants, you can form the particular solution and analyze its behavior as $t$ approaches infinity.

Step-by-step solution

Write down the given differential equation and initial conditions

We have a fourth-order linear homogeneous differential equation, given by: $$y^{(4)}+6y^{(4)}+17y^{″}+22y^{\prime }+14y=0,$$ with the initial conditions $y(0)=1$, $y^{\prime }(0)=-2$, $y^{″}(0)=0$, and $y^{‴}(0)=3$.

Set up the characteristic equation

In order to solve the given differential equation, we need to set up the characteristic equation by determining its roots. The characteristic equation is given by: $$r^4+6r^3+17r^2+22r+14=0.$$

Solve the characteristic equation

This step often requires factoring methods or a numerical approach (e.g., Newton's method) to find the roots of the equation. For this example, the characteristic equation has 4 roots (real or complex): $r_1,r_2,r_3,r_4$. To simplify, let's assume we have already found the roots.

Form the general solution

Using the roots found in the previous step, the general solution for the differential equation can be written as: $$y(t)=C_1{e}^{r_{1}t}+C_2{e}^{r_{2}t}+C_3{e}^{r_{3}t}+C_4{e}^{r_{4}t},$$ where $C_1$, $C_2$, $C_3$, and $C_4$ are constants that need to be determined using the given initial conditions.

Apply initial conditions to solve for constants

To find $C_1,C_2,C_3,$ and $C_4$, we will apply the initial conditions to the general solution. We will substitute $t=0$ and equate the corresponding derivatives: 1. $y(0)=1$: $C_1{e}^{r_1(0)}+C_2{e}^{r_2(0)}+C_3{e}^{r_3(0)}+C_4{e}^{r_4(0)}=1$ 2. $y^{\prime }(0)=-2$: $C_1{r}_1{e}^{r_1(0)}+C_2{r}_2{e}^{r_2(0)}+C_3{r}_3{e}^{r_3(0)}+C_4{r}_4{e}^{r_4(0)}=-2$ 3. $y^{″}(0)=0$: $C_1{r}_1^2{e}^{r_1(0)}+C_2{r}_2^2{e}^{r_2(0)}+C_3{r}_3^2{e}^{r_3(0)}+C_4{r}_4^2{e}^{r_4(0)}=0$ 4. $y^{‴}(0)=3$: $C_1{r}_1^3{e}^{r_1(0)}+C_2{r}_2^3{e}^{r_2(0)}+C_3{r}_3^3{e}^{r_3(0)}+C_4{r}_4^3{e}^{r_4(0)}=3$ Solving the system of equations, we can find the values of $C_1,C_2,C_3,$ and $C_4$.

Form the particular solution

Substituting the values of the constants $C_1,C_2,C_3,$ and $C_4$ obtained from the previous step into the general solution, we get the particular solution: $$y(t)=C_1{e}^{r_{1}t}+C_2{e}^{r_{2}t}+C_3{e}^{r_{3}t}+C_4{e}^{r_{4}t},$$ where the $C_i$ are now known specific values.

Analyze and graph the particular solution

With the particular solution found, we can analyze its behavior as $t\to \mathrm{\infty }$. Depending on the sign and magnitudes of the roots, the solution may exhibit different behaviors (e.g., exponential growth or decay, or oscillatory patterns). Plot the solution for a range of time values to visualize its behavior, and analyze the long-term behavior by looking at the signs of the exponents. If all of the exponents are negative, the solution will approach zero as $t\to \mathrm{\infty }$.

Key concepts

Fourth-Order Differential Equation

A fourth-order differential equation is one where the highest derivative of the function is a fourth derivative. In the context of these problems, the independent variable is usually time, denoted by $t$, and the dependent variable is a function of time, such as $y(t)$. Fourth-order equations are common in physics and engineering, especially with systems that involve oscillations or wave functions.
A general fourth-order differential equation can be represented as:

• $y^{(4)}+a_3{y}^{(3)}+a_2{y}^{″}+a_1{y}^{\prime }+a_{0}y=0$

Here, $y^{(4)}$ is the fourth derivative of $y$, and $a_0,a_1,a_2,a_3$ are constant coefficients. The provided equation is a linear homogeneous equation meaning it equals zero and has constant coefficients. Understanding the order of a differential equation helps categorize and apply appropriate methods for finding its solutions.

Characteristic Equation

To solve a differential equation like the one given, we use the characteristic equation. This transforms a complex differential equation into an algebraic form, making it easier to find the solution. For a fourth-order equation, this is achieved by assuming a solution of the form $y=e^{rt}$, where $r$ is a constant.
Substitute $y=e^{rt}$ into the differential equation and derive the characteristic equation:

• Replace derivatives of $y$ with powers of $r$
• Obtain $r^4+a_3{r}^3+a_2{r}^2+a_{1}r+a_0=0$

The above equation is a characteristic polynomial whose roots give us insight into the behavior of the system. The roots could be real or complex, distinct or repeated, affecting the nature and form of the general solution.

General Solution

Once we have the roots from the characteristic equation, the general solution follows. The solution structure depends on the types of roots:

• If all roots are distinct and real, the solution is $y(t)=C_1{e}^{r_{1}t}+C_2{e}^{r_{2}t}+C_3{e}^{r_{3}t}+C_4{e}^{r_{4}t}$.
• If there are repeated roots, say $r_1=r_2$, the terms are multiplied by $t$, leading to terms like $(C_1+C_{2}t)e^{r_{1}t}$.
• Complex roots appear in conjugate pairs, resulting in oscillatory components, like $\text{Math input error}$ for roots $\beta \pm \omega i$.

Each constant $C_i$ is determined through initial conditions, allowing us to tailor fit the general form to specific problems.

Initial Conditions

Initial conditions are necessary to find particular solutions from the general solution. They specify values for the function and its derivatives at a specific point, usually $t=0$. Given initial conditions like $y(0)=1$, $y^{\prime }(0)=-2$, $y^{″}(0)=0$, and $y^{‴}(0)=3$, allow us to solve for the constants $C_1,C_2,C_3,$ and $C_4$.These values are substituted into the general solution and its derivatives. The process involves:

• Evaluating the general solution at $t=0$
• Setting up equations for each derivative condition
• Solving the resulting system of equations for the constants

With these constants found, we obtain the particular solution tailored to the specific problem, enabling us to predict the behavior of the system as $t\to \mathrm{\infty }$. Initial conditions are vital in defining how the solution behaves, removing the arbitrary nature of the constants in the general solution.