Question

Graph several functions that satisfy each of the following differential equations. Then find and graph he particular fimction that satisfies the given initial condition. $$f^{\prime}(x)=3 x^{2}-1 ; f(1)=2$$

Short answer

Answer: The particular solution of the given differential equation with the given initial condition is $$f(x)=x^3-x+2$$.

Step-by-step solution

Find the general solution

In order to find the general solution, we need to integrate the given differential equation: $$f^{\prime}(x)=3x^2-1$$ Integrate both sides with respect to x: $$\int{f^{\prime}(x)\,dx} = \int{(3x^2-1)\,dx}$$ $$f(x)=\int{(3x^2-1)\,dx}=x^3-x+C$$ Now we have the general solution of the given differential equation: $$f(x)=x^3-x+C$$ where C is the constant of integration.

Apply the initial condition

Now we will apply the initial condition: $$f(1)=2$$ Plug in the initial condition: $$2=(1)^3-(1)+C$$ Solve for the constant C: $$C=2$$

Find the particular solution

Now that we have the value of C, we can find the particular solution: $$f(x)=x^3-x+2$$

Graph the family of functions satisfying the differential equation

Plot the family of functions satisfying the given differential equation using the general solution: $$f(x)=x^3-x+C$$ We can plot a few cases (C = -1, 0, 1, 3) to show the family of functions that satisfy the differential equation. Remember that any value for C will result in a function that satisfies the differential equation.

Graph the particular function satisfying the initial condition

Now, plot the particular solution with the initial condition: $$f(x)=x^3-x+2$$ This graph has a unique curve, as it is the only function from the family of solutions that passes through the point (1, 2). To recap, we found the general solution of the given differential equation, $$f(x)=x^3-x+C$$, and after applying the initial condition, found the particular solution as $$f(x)=x^3-x+2$$. We graphed the family of functions that satisfy the differential equation and the particular function that satisfies the given initial condition.

Key concepts

Integration

Integration is a fundamental technique in calculus used to find a function when its derivative is known. In this problem, the differential equation given is \(f^{\prime}(x) = 3x^2 - 1\). Integration will help us determine the original function, \(f(x)\), from this derivative. By integrating both sides of the equation with respect to \(x\), we find:

• \(f(x) = \int{(3x^2 - 1)\, dx} = x^3 - x + C\)

Here, \(C\) is the constant of integration. This constant is crucial because it represents any constant that would disappear upon taking the derivative of \(f(x)\). Hence, integration is the process of working backwards from the derivative to the original function.

Initial Conditions

Initial conditions provide specific values that allow us to solve for the constant of integration \(C\) in the general solution of a differential equation. In the given problem, the initial condition is stated as \(f(1) = 2\). This means that when \(x = 1\), the function \(f(x)\) outputs the value 2. To apply this condition:

• Substitute \(x = 1\) into the general solution: \(f(1) = 1^3 - 1 + C = 2\).
• Solve for \(C\): \(1 - 1 + C = 2\) simplifies to \(C = 2\).

This step transforms the general solution into a particular solution that satisfies the exact given conditions, making it unique for that specific scenario.

Particular Solution

The particular solution is a specific solution to the differential equation that satisfies the given initial conditions. Once the constant \(C\) is determined using the initial conditions, the function becomes a particular solution. From our problem:

• General solution: \(f(x) = x^3 - x + C\)
• With \(C = 2\), the particular solution is: \(f(x) = x^3 - x + 2\)

This particular solution not only solves the differential equation but also passes through the initial point given in the problem, which is \((1, 2)\). It effectively narrows down the infinite family of solutions to just one specific solution.

Graphing Functions

Graphing functions involves plotting points in a coordinate plane to visualize the behavior of a function. For our differential equation, you first graph the family of functions described by the general solution \(f(x) = x^3 - x + C\), where \(C\) can take various values.

• Use different values of \(C\) (for example, \(C = -1, 0, 1, 2\)) to see how they shape different curves on the graph.
• This portrays the infinite variety of solutions depending on the constant \(C\).

The next step is to draw the graph of the particular solution \(f(x) = x^3 - x + 2\). This graph will cross through the point \((1, 2)\), making it unique among the family of curves. By visualizing these graphs, you better understand the relationship between the general and particular solutions and see how the initial condition affects the function's shape.