Question

Graph several functions that satisfy the following differential equations. Then find and graph the particular function that satisfies the given initial condition. $$f^{\prime}(x)=3 x+\sin \pi x ; f(2)=3$$

Short answer

Question: Find the particular function that satisfies the differential equation \(f^{\prime}(x) = 3x + \sin(\pi x)\) and the initial condition \(f(2)=3\). Answer: The particular function that satisfies the given differential equation and initial condition is: $$f(x) = \frac{3}{2}x^2 - \frac{1}{\pi}\cos(\pi x) + \frac{1}{\pi} - 3.$$

Step-by-step solution

Integration of the differential equation

Given the differential equation: $$f^{\prime}(x) = 3x + \sin(\pi x)$$ We need to find f(x). We integrate both sides of the equation with respect to x: $$\int f^{\prime}(x) dx = \int (3x + \sin(\pi x)) dx$$ Integrating each term, we get: $$f(x) = \int 3x dx + \int \sin(\pi x) dx$$ Now, we'll find the indefinite integral of each term.

Integral of the first term

To find the integral of the first term (3x), we use: $$\int 3x dx = \frac{3}{2}x^2 + C_1$$ Here, C_1 is a constant.

Integral of the second term

To find the integral of the second term (\(\sin(\pi x)\)), we use: $$\int \sin(\pi x) dx = -\frac{1}{\pi}\cos(\pi x) + C_2$$ Here, C_2 is another constant.

Obtain the general solution

Based on Steps 2 and 3, we now write the general solution of the given differential equation as: $$f(x) = \frac{3}{2}x^2 - \frac{1}{\pi}\cos(\pi x) + C$$ where $$C = C_1 + C_2$$ is the constant of integration.

Use the initial condition

Now, we must find the particular solution that satisfies the initial condition \(f(2)=3\). To do that, we substitute x with 2 in the general solution derived in step 4: $$3 = \frac{3}{2}(2)^2 - \frac{1}{\pi}\cos(\pi (2)) + C$$ Plugging the values, we get: $$3 = 6 - \frac{1}{\pi}\cos(2\pi) + C$$ Since \(\cos(2\pi) = 1\), the equation becomes: $$3 = 6 - \frac{1}{\pi} + C$$ Now we can solve for C: $$C = \frac{1}{\pi} - 3$$

Find the particular solution

Now that we have the value of the constant C, we can write the particular solution: $$f(x) = \frac{3}{2}x^2 - \frac{1}{\pi}\cos(\pi x) + \frac{1}{\pi} - 3$$ This particular function satisfies both the given differential equation and the initial condition.

Key concepts

Integration

When we start solving differential equations, integration plays a vital role. In this exercise, we're dealing with a first derivative, so we use integration to find the original function, often referred to as the antiderivative.

The differential equation given is \(f'(x) = 3x + \sin(\pi x)\). The task is to find \(f(x)\). We do this by integrating both sides:

• \(\int f'(x) dx = \int (3x + \sin(\pi x)) dx\)

Integration is performed term by term. For each term:

• \(\int 3x dx\) yields \(\frac{3}{2}x^2 + C_1\) (using the power rule)


• \(\int \sin(\pi x) dx\) results in \(-\frac{1}{\pi}\cos(\pi x) + C_2\) (employing trigonometric integration techniques)

Combining these, we have the general solution:
\[f(x) = \frac{3}{2}x^2 - \frac{1}{\pi}\cos(\pi x) + C\]

The constant \(C\) represents the sum of the constants \(C_1\) and \(C_2\). This general form is crucial in handling different initial conditions, leading us to the next concept.

Initial Condition

Initial conditions are crucial because they allow us to find a specific solution from a general one. In other words, they help us pin down one unique function fitting the context.

We have an initial condition given as \(f(2)=3\). This tells us that the function \(f\) takes the value 3 when \(x\) equals 2. Given the general solution:
\[f(x) = \frac{3}{2}x^2 - \frac{1}{\pi}\cos(\pi x) + C\]
we use \(x = 2\) to find \(C\):

• Substitute \(x=2\) in the equation.


• \(3 = \frac{3}{2}(2)^2 - \frac{1}{\pi}\cos(2\pi) + C\)

Since \(\cos(2\pi) = 1\), it simplifies further, enabling us to solve for \(C\):
\[C = \frac{1}{\pi} - 3\]

By solving this, we ascertain the specific constant needed to satisfy the initial condition, thus narrowing down the infinite general solutions to our required particular solution.

Particular Solution

Once we have determined the constant\(C\), the final step is writing the particular solution that uniquely satisfies both the differential equation and the initial condition.

Plugging \(C = \frac{1}{\pi} - 3\) back into the general solution gives us the particular solution:
\[f(x) = \frac{3}{2}x^2 - \frac{1}{\pi}\cos(\pi x) + \frac{1}{\pi} - 3\]

This function represents the specific behavior of the system as described by the differential equation \(f'(x) = 3x + \sin(\pi x)\) and the initial value \(f(2) = 3\).

The particular solution is essential as it pinpoints the exact curve out of a family of potential solutions, precisely meeting the criteria laid out by the initial condition.