Question

Find the solution of the initial value problem $$ m y^{\prime \prime}+c y^{\prime}+k y=0, \quad y(0)=y_{0}, y^{\prime}(0)=v_{0} $$ given that the motion is overdamped, so the general solution of the equation is $$ y=c_{1} e^{r_{1} t}+c_{2} e^{r_{2} t}\left(r_{1}, r_{2}<0\right) $$

Short answer

Answer: The final solution is $$y(t) = \frac{v_0 - y_0 r_2}{r_1-r_2} e^{r_1 t} + \left(y_0 - \frac{v_0 - y_0 r_2}{r_1-r_2}\right) e^{r_2 t}$$.

Step-by-step solution

Identify initial conditions

We are given the initial conditions: \(y(0) = y_0\) and \(y'(0) = v_0\).

Apply the first initial condition

Plug in \(t = 0\) and \(y(0) = y_0\) into the general solution: $$y_0 = c_1 e^{r_1 \cdot 0} + c_2 e^{r_2 \cdot 0}$$ Since \(e^0 = 1\), we have \(y_0 = c_1 + c_2\).

Calculate the derivative of the general solution

Differentiate the general solution with respect to \(t\) to find \(y'(t)\): $$y'(t)=c_1 r_1 e^{r_1 t}+c_2 r_2 e^{r_2 t}$$

Apply the second initial condition

Plug in \(t = 0\) and \(y'(0) = v_0\) into the derivative of the general solution: $$v_0 = c_1 r_1 e^{r_1\cdot 0} + c_2 r_2 e^{r_2\cdot 0}$$ Since \(e^0 = 1\), we have \(v_0 = c_1 r_1 + c_2 r_2\).

Solve the system of linear equations

We now need to solve the system of linear equations to find \(c_1\) and \(c_2\): $$ \begin{cases} c_1 + c_2 = y_0 \\ c_1 r_1 + c_2 r_2 = v_0 \end{cases} $$ This can be done using substitution or elimination method, among others. We will use substitution here. Solve the first equation for \(c_2\): $$c_2 = y_0 - c_1$$ Now, substitute this expression for \(c_2\) into the second equation: $$c_1 r_1 + (y_0 - c_1)r_2 = v_0$$

Solve for \(c_1\)

Distribute \(r_2\) and collect \(c_1\) terms: $$c_1 (r_1-r_2) = v_0 - y_0 r_2$$ Now, solve for \(c_1\): $$c_1 = \frac{v_0 - y_0 r_2}{r_1-r_2}$$

Solve for \(c_2\)

Substitute the found value of \(c_1\) back into the expression for \(c_2\): $$c_2 = y_0 - \frac{v_0 - y_0 r_2}{r_1-r_2}$$

Write the final solution

Plug the values of \(c_1\) and \(c_2\) back into the general solution: $$y(t) = \frac{v_0 - y_0 r_2}{r_1-r_2} e^{r_1 t} + \left(y_0 - \frac{v_0 - y_0 r_2}{r_1-r_2}\right) e^{r_2 t}$$ This is the solution to the initial value problem for the given overdamped motion equation.

Key concepts

Overdamped Motion

When learning about the motion of objects, an important concept to understand is overdamped motion, which occurs in systems where the damping force is higher than the critical damping required to prevent oscillations. This means that the system returns to equilibrium without oscillating around the final position.

In the context of the given initial value problem, overdamped motion is represented by a second-order linear differential equation where the coefficients indicate a strong damping effect. This is mathematically expressed as a characteristic equation with two real, distinct, and negative roots, denoted by r1 and r2. As the roots are negative, the solution to the differential equation reflects exponential decay of the motion over time.

The general solution for an overdamped system is a sum of two exponentials, which incorporate the initial conditions of the system to adapt the solution to specific starting parameters, such as initial displacement y0 and initial velocity v0. Understanding the behavior of overdamped systems is crucial for many practical engineering applications, such as automotive shock absorbers and building dampers that prevent excessive swaying during earthquakes.

Differential Equations

Differential equations are powerful mathematical tools that describe the relationship between a function and its derivatives, representing how a particular quantity changes over time. They can be used to model a wide range of phenomena in engineering, physics, economics, and beyond.

The initial value problem presented involves a second-order differential equation my'' + cy' + ky = 0, where m, c, and k are constants. The notation y'' denotes the second derivative of y with respect to time, which represents acceleration in the physical system being modeled. Meanwhile, y' represents velocity. The key to solving an initial value problem is to integrate the differential equation while incorporating the given initial conditions to produce a unique solution.

For this problem, we employed the particular solution for overdamped motion—a linear combination of exponential functions with coefficients determined by the initial conditions. By substituting these conditions into the solution and its derivative, we achieved a system of linear equations to solve for unknown coefficients. In essence, mastering differential equations enables one to predict system behavior under various scenarios.

Linear Equations

Linear equations form the bedrock of algebra and provide an essential foundation for solving more complex mathematical problems. A linear equation is an equation where each term is either a constant or the product of a constant and a single variable, and no variable is raised to a power greater than one. A system of linear equations is a set of two or more linear equations involving the same set of variables.

Substitution and Elimination Methods

In the step-by-step solution, we encounter a system of linear equations with two unknowns, c1 and c2. To solve this system, substitution or elimination methods are often used. These techniques reorganize the equations to isolate one variable at a time, making it possible to determine the values of both unknowns systematically.

These linear equations directly influence the coefficients of the exponential functions in the general solution, ensuring that both the motion condition and the initial velocities are satisfied. A firm understanding of how to manipulate and solve systems of linear equations is critical for students as it applies to countless situations across mathematics and science.