Question

Find the particular solution to the differential equation \(8 \frac{d x}{d t}=-2 \cos (2 t)-\cos (4 t)\) that passes through \((\pi, \pi)\), given that \(x=C-\frac{1}{8} \sin (2 t)-\frac{1}{32} \sin (4 t)\) is a general solution.

Short answer

The particular solution is \(x = \pi - \frac{1}{8} \sin(2t) - \frac{1}{32} \sin(4t)\).

Step-by-step solution

Apply Initial Condition

Start by using the initial condition to find the value of the constant C in the general solution. We have the general solution: \[x = C - \frac{1}{8} \sin (2t) - \frac{1}{32} \sin (4t)\]The initial condition given is \(x(\pi) = \pi\), so let's substitute \(t = \pi\) and \(x = \pi\) into the equation.Substitute:\[\pi = C - \frac{1}{8} \sin (2\pi) - \frac{1}{32} \sin (4\pi)\]

Evaluate Sine Functions

Evaluate the sine functions at \(t = \pi\):\(\sin(2\pi) = 0\) and \(\sin(4\pi) = 0\) because sine of any integer multiple of \(\pi\) is zero.Substitute these into the equation:\[\pi = C - \frac{1}{8} \times 0 - \frac{1}{32} \times 0\]This simplifies to:\[\pi = C\]

Solve for C

From the simplified equation:\[\pi = C\]We find that \(C = \pi\).

Write the Particular Solution

Substitute the value of \(C\) back into the general solution to get the particular solution of the differential equation:\[x = \pi - \frac{1}{8} \sin(2t) - \frac{1}{32} \sin(4t)\]

Key concepts

Particular Solution

A particular solution is a specific solution to a differential equation that satisfies an initial condition. For the differential equation we have, finding a particular solution involves determining a constant that meets specific criteria or initial condition given. In this case, the differential equation is \(8 \frac{d x}{d t} = -2 \cos (2 t) - \cos (4 t)\), and we know its general solution is \(x = C - \frac{1}{8} \sin (2t) - \frac{1}{32} \sin (4t)\).
To find the particular solution, we use an initial condition, which is a known value of the solution at a certain point. Here, the given initial condition is \(x(\pi) = \pi\). By substituting this into the general solution, we solve for the constant \(C\) and find the particular solution that satisfies this differential equation and passes through the point \( (\pi, \pi) \). The final particular solution in this example is \(x = \pi - \frac{1}{8} \sin(2t) - \frac{1}{32} \sin(4t)\).
This solution is a necessary step because it connects the general behavior of the differential equation with specific conditions that occur in practical applications.

Initial Condition

Initial conditions are crucial in solving differential equations because they provide specific values that the solution must satisfy. An initial condition gives us information about the behavior of the solution at a certain point, usually at the start of the problems timeline. Here, the initial condition is \(x(\pi) = \pi\).
Applying an initial condition is an essential process when determining a particular solution from a general solution. It involves substituting the known point values into the general solution to calculate the unknown constant. In this example, by substituting \(t = \pi\) and \(x = \pi\) into the general solution, we work out that \(C = \pi\).
This calculation ensures that the differential equation trajectory goes through the specified coordinate, giving us the tailored solution needed for specific problem scenarios. Initial conditions help anchor the general theoretical solution to solutions likely to be encountered in real-world scenarios.

General Solution

The general solution of a differential equation provides a formula that includes a constant, allowing for multiple specific solutions. In other words, it represents an entire family of functions that satisfy the differential equation. For the given differential equation, the general solution is \(x = C - \frac{1}{8} \sin (2t) - \frac{1}{32} \sin (4t)\).
The presence of the constant \(C\) reflects the infinite number of solutions corresponding to different initial conditions. By itself, it doesn’t cater to any specific real-world situation until further conditions are applied. Using initial conditions, one of these potential solutions is selected to serve as the particular solution.
In mathematical modeling, the general solution serves as a broad template from which specific solutions can be derived when you apply particular conditions. This way, the general solution provides a comprehensive understanding and flexibility needed for a wide range of situations.

Sine Function Evaluation

Sine function evaluation plays a key role in solving trigonometric differential equations. The sine function, denoted as \(\sin(t)\), is periodic with specific characteristics at certain points. In this exercise, we utilize the property that \(\sin(n\pi) = 0\), where \(n\) is any integer, for evaluating trigonometric functions in the general solution.
For this differential equation, evaluating \(\sin(2\pi)\) and \(\sin(4\pi)\) at the initial condition was necessary to simplify the equation. Both evaluate to zero because they are sine functions at integer multiples of \(\pi\), which is a known angle where the sine value is zero. Substituting these values into the equation \(\pi = C - \frac{1}{8} \times 0 - \frac{1}{32} \times 0\) leads directly to \(\pi = C\).
Understanding how sine and other trigonometric functions behave at crucial points allows for simplification of differential equations and efficient solution finding. It is an essential skill in solving many mathematical and physics-related problems where waveform functions play a role.