Question

If \(\phi(x)=\cos x-\int_{0}^{x}(x-t) \phi(t) d t\), then \(\phi^{\prime \prime}(x)+\phi(x)\) is equal to (1) \(-\cos x\) (2) 0 (3) \(\int_{0}^{x}(x-t) \phi(t) d t\) (4) \(-\int_{0}^{-x}(x-t) \phi(t) d t\) (5) \(\cos x\)

Short answer

The correct answer is (1) \(\-cos x\).

Step-by-step solution

Differentiate \(\frac{d}{dx} \phi(x) \)

Find the first derivative of \(\phi(x) = \cos x - \int_{0}^{x} (x - t) \phi(t) dt\). Use the Leibniz rule for differentiation under the integral sign. \[\phi'(x) = -\sin x - \int_{0}^{x} \phi(t) dt\]

Differentiate the First Derivative

Now, differentiate \(\frac{d}{dx} \phi'(x) \) to find the second derivative. \[\phi''(x) = -\cos x + \phi(x) \]

Combine the Results

Combine \(\frac{d^{2}{dx^{2}} \phi(x) \) and \(\phi(x) \) to verify the expression. \[\phi''(x) + \phi(x) = -\cos x + \phi(x) + \phi(x) = -\cos x\]

Identify the Correct Option

Compare the result with the options given. The expression \(\phi''(x) + \phi(x) == -\cos x \) matches option (1).

Key concepts

Differential Equations

Differential equations are mathematical equations that relate a function with its derivatives. They show how a function changes and provide a way to model real-world phenomena, like physics, biology, and engineering problems. In simpler terms, they tell us how one thing changes in relation to another. For example, if you're studying the speed of a falling object, a differential equation can help explain how the speed changes over time.
In our exercise, \(\frac{d}{dx} \phi(x)\) and \(\frac{d^{2}{dx^{2}} \phi(x)\) are examples of differential equations. To solve these equations, we need to know both the function and its derivatives. By solving differential equations, we find the function (\phi(x) in this case) that satisfies the relationship described between the function and its derivatives.

Leibniz Rule

The Leibniz rule is essential for differentiating under the integral sign. This rule is particularly useful when dealing with integrals that have variable limits of integration. It states that if you have a function defined as an integral, you can differentiate it with respect to a variable. In other words, if we define a function \( \phi(x) = \cos x - \int_{0}^{x} (x - t) \phi(t) \), we can use the Leibniz rule to find its derivative.
In our exercise, we used the Leibniz rule in Step 1 to find the first derivative of \( \phi(x)\). Here's a simplified version of how the rule works:
\( \frac{d}{dx} \int_{a(x)}^{b(x)} f(x,t) dt = f(x, b(x)) \frac{d}{dx} b(x) - f(x, a(x)) \frac{d}{dx} a(x) + \int_{a(x)}^{b(x)} \frac{\backslashpartial}{\backslashpartial x} f(x,t) dt \)
In our case, it helped us to differentiate and make sense of the expression involving the integral. This is a powerful tool because it simplifies complex problems by breaking them into more manageable pieces.

Integration

Integration is the process of finding the integral of a function, which can be thought of as the reverse of differentiation. It helps compute areas under curves, accumulated quantities, and many other useful properties. There are two main types of integrals:

• Definite Integrals: They have upper and lower limits of integration and provide a numerical value for the area under a curve.
• Indefinite Integrals: These do not have limits and represent a family of functions, including a constant of integration.

In our exercise, we integrated the function \( (x - t) \phi(t) \) from 0 to x. This integral is part of the definition of \(\phi(x) \). Specifically, we calculated:
\( \int_{0}^{x} (x - t) \phi(t) dt \)
Understanding integration allows us to solve more advanced problems in calculus, like the one in our exercise. By mastering both differentiation and integration, you can tackle a wide range of mathematical challenges and apply these techniques in various fields, such as physics, engineering, and economics.