Problem 46 - Calculus: Early Transcendental Functions

Question

Find all functions satisfying the given conditions. $$f^{\prime \prime \prime}(x)=\sin x-e^{x}$$

Step-by-Step Solution

Answer

The solutions to the given differential equation are all functions of the form \(f(x) = \cos x - e^{x} + \frac{1}{2}C_1x^2 + C_2x + C_3\), where \(C_1\), \(C_2\), and \(C_3\) can be any real numbers.

Step 1: First Integration

First, integrate the given differential equation once: \(\int (f^{\prime \prime \prime}(x))dx = \int (\sin x - e^{x})dx\). This results in \(f^{\prime \prime}(x)= -\cos x - e^{x} + C_1\), where \(C_1\) is the constant of integration.

Step 2: Second Integration

Now integrate the obtained expression from step 1: \(\int (f^{\prime \prime}(x))dx = \int (-\cos x - e^{x} + C_1)dx\). This results in \(f^{\prime}(x) = -\sin x - e^{x} + C_1x + C_2\), where \(C_2\) is the second constant of integration.

Step 3: Third Integration

Now, integrate the result from step 2 one more time: \(\int (f^{\prime}(x))dx = \int (-\sin x - e^{x} + C_1x + C_2)dx\). This gives \(f(x) = \cos x - e^{x} + \frac{1}{2}C_1x^2 + C_2x + C_3\), where \(C_3\) is the third constant of integration.

Key Concepts

Integration

Integration is a fundamental concept in calculus that involves finding the antiderivative of a function. It is often used to determine accumulated quantities, such as area under a curve or to reconstruct a function from its derivative. In the process of solving differential equations, integration allows us to "reverse" the derivation process.

When you integrate a function, you essentially find a new function whose derivative gives you the original function. This is crucial in problems involving derivatives and differentials, where the aim is to determine the original function from its rate of change.

For example, in the exercise above, the differential equation given is a third derivative, and integration is applied three times to get back to the original function form, adding constants of integration after each step.

Constant of Integration

The constant of integration appears when we integrate a function, because integration is an inverse process to differentiation. Differentiation removes constant terms, so when we integrate, we must account for all potential constant values with an arbitrary constant, often denoted as C.

• When you integrate a function once, you introduce one constant of integration, represented as \(C_1\). For instance, when you integrate \(-\cos x - e^{x}\), it becomes \(-\cos x - e^{x} + C_1\).

• Subsequent integrations introduce additional constants. In the solution above, each integration step produces a new constant: \(C_2\) after the second integration, and \(C_3\) after the third.

These constants represent unknown initial conditions that can be determined if further information about the function is given, such as specific values of the original function at particular points.

Calculus

Calculus, often seen as the mathematics of change, comprises two main branches: differential calculus and integral calculus.

Differential calculus focuses on rates of change and slopes of curves by utilizing derivatives. It zeroes in on how the function's output changes as its input changes and is incredibly useful in fields like physics, engineering, and economics.

Integral calculus, on the other hand, deals with the accumulation of quantities and areas under curves through integration. It essentially allows us to reconstruct a function from its derivative, as depicted in the exercise's solution process.

Through calculus, we are equipped with the tools to model and solve real-world problems involving changing variables and systems. Understanding both differentiation and integration is key for analyzing and predicting behaviors within a mathematical framework.