Problem 41 - Essential Calculus

Question

Solve the differential equation. $$ f^{\prime}(x)=4 x, f(0)=6 $$

Step-by-Step Solution

Answer

The solution to the given differential equation is \(f(x) = 2x^2 + 6\).

Step 1: Integrate the differential equation

Here the derivative of a function, \(f'(x) = 4x\), is given. It is the rate of change of the function. To find the original function \(f(x)\), we have to integrate the given function. So, integrate \(f'(x)\) with respect to \(x\). This gives us the integral,\[\int f'(x) \, dx = \int 4x \, dx = 2x^2 + C\]where C is the constant of integration.

Step 2: Apply the initial condition to find the constant of integration

The initial condition given is \(f(0) = 6\). This means when \(x = 0\), \(f(x) = 6\). Substitute these values into the integrated function we obtained in step 1:\(2(0)^2 + C = 6\) , which simplifies to \(C = 6\) .

Step 3: Write the final solution

Now, substitute the obtained value of C into the integrated function. This gives us the final solution:\[f(x) = 2x^2 + 6\]This is the solution to the differential equation.

Key Concepts

Integration

Integration is at the heart of solving differential equations. It's a process used to determine a function whose derivative is the provided function. Think of it like reverse-engineering a problem: you are given the rate of change of something, and you need to find the original amount.

• In the given differential equation, the derivative is \( f'(x) = 4x \).
• Through integration, we aim to find \( f(x) \), the original function.

To perform the integration, we use the power rule for integrals. The integral of \( 4x \) with respect to \( x \) is \( 2x^2 \), but remember, integration without limits results in an arbitrary constant \( C \). This constant accounts for all possible vertical shifts of the function on the graph. Hence, the integral part becomes \( \int 4x \, dx = 2x^2 + C \). The concept of a constant being added underlines the idea that indefinite integrals represent a family of curves. The next step is to discover which of these curves matches our specific case, utilizing initial conditions.

Initial Conditions

Initial conditions solve a crucial part of the puzzle when integrating differential equations. They help determine the specific solution from an infinite set of possibilities presented by the integration process.

• In our original exercise, the initial condition is \( f(0) = 6 \).
• This means that when \( x = 0 \), the function \( f(x) \) should equal 6.

By utilizing the initial condition, we substitute \( x = 0 \) into the integrated result \( 2x^2 + C \) and set it equal to the initial value. Thus, \( 2(0)^2 + C = 6 \) simplifies to \( C = 6 \). This process validates that the specific solution to our problem is not just any function, but precisely one that passes through the point \( (0, 6) \) in the graph. Initial conditions are vital because they ensure that the solution one finds is not just correct mathematically but also fits the real-world scenario being modeled.

Constant of Integration

The constant of integration \( C \) emerges when you perform an indefinite integral. It represents any constant value that could have existed in the original function before differentiation. All anti-derivatives of a function differ by such a constant.

• During integration of the given function \( f'(x) = 4x \), we encountered \( C \) when expressing \( 2x^2 + C \).
• Without additional information like an initial condition, \( C \) would remain undetermined.

In the context of our example, we utilized the initial condition \( f(0) = 6 \) to determine the value of \( C \). This pinpointed our equation to precisely describe \( f(x) = 2x^2 + 6 \). Remember that while \( C \) can be any real number, its exact value gives the unique solution that aligns the theoretical model with observational data or specified conditions. Thus, the constant of integration is more than just a mathematical artifact; it's a bridge to align equations with physical phenomena or specific cases in calculus problems.