{Use of Tech} Graphing general solutions Graph several functions that satisfy each of the following differential equations. Then find and graph the particular function that satisfies the given initial condition.


f'(x) = 3x + sinx; f(0) = 3

Verified step by step guidance

1. Step 1: Understand the problem. The given differential equation is f'(x) = 3x + sin(x), and the initial condition is f(0) = 3. This means we need to find the general solution to the differential equation and then determine the particular solution that satisfies the initial condition.
2. Step 2: Integrate the given differential equation to find the general solution. Recall that the derivative f'(x) represents the rate of change of f(x). To find f(x), integrate the right-hand side of the equation: ∫(3x + sin(x)) dx. Break this into two separate integrals: ∫3x dx and ∫sin(x) dx.
3. Step 3: Solve each integral. For ∫3x dx, use the power rule: ∫x^n dx = (x^(n+1))/(n+1). For ∫sin(x) dx, recall the standard integral: ∫sin(x) dx = -cos(x). Combine these results to form the general solution: f(x) = (3/2)x^2 - cos(x) + C, where C is the constant of integration.
4. Step 4: Apply the initial condition f(0) = 3 to determine the value of C. Substitute x = 0 into the general solution: f(0) = (3/2)(0)^2 - cos(0) + C. Simplify this expression to solve for C, noting that cos(0) = 1.
5. Step 5: Write the particular solution using the value of C obtained in Step 4. Then, use graphing technology (such as a graphing calculator or software) to plot the general solution f(x) = (3/2)x^2 - cos(x) + C for various values of C, and the particular solution that satisfies f(0) = 3.

Key Concepts

Differential Equations

A differential equation is an equation that relates a function to its derivatives. In this case, the equation f'(x) = 3x + sin(x) describes how the function f(x) changes with respect to x. Solving this equation involves finding a function whose derivative matches the given expression.

General Solution

The general solution of a differential equation includes all possible solutions and typically contains arbitrary constants. For the given equation, integrating f'(x) will yield a general solution that includes a constant of integration, representing a family of functions that satisfy the differential equation.

Initial Conditions

Initial conditions are specific values that a solution must satisfy at a given point, which helps to determine the particular solution from the general solution. In this case, the condition f(0) = 3 allows us to find the specific value of the constant in the general solution, ensuring that the function passes through the point (0, 3) on the graph.