Question

Determining a Solution In Exercises \(13-20\) , determine whether the function is a solution of the differential equation \(y^{(4)}-16 y=0\) . $$ y=3 \cos 2 x $$

Short answer

Yes, the function \(y=3 \cos 2x\) is a solution to the differential equation \(y^{(4)} -16y=0\).

Step-by-step solution

Compute the 4th derivative of the function

The 4th order derivative of the function \(y = 3 \cos 2x\) should be calculated. Using the chain rule and understanding the periodic property of cosine functions, it can be computed as: \(y^{(4)} = 3 \cdot (2)^{4} \cdot \cos 2x\).

Substitute the function and its derivative into the equation

Now, substitute the values of \(y\) and \(y^{(4)}\) into the differential equation. The resulting equation after substitution is: \(48 \cos 2x - 16 \cdot 3 \cos 2x = 0\)

Verify whether the equation holds true

The equation simplifies to \(48 \cos 2x - 48 \cos 2x = 0\), which is obviously true for all values of \(x\). Thus, the function \(y = 3\cos2x\) is indeed a solution to the given differential equation.

Key concepts

Higher-Order Derivatives

Understanding higher-order derivatives is crucial when dealing with complex differential equations. In simple terms, a higher-order derivative represents the rate of change of a function's rate of change. For instance, if the first derivative represents velocity, the second derivative represents acceleration. As you go higher in order, each derivative represents the change in the previous derivative.

When solving differential equations, particularly those of fourth order and above, it requires proficiency in taking multiple derivatives of a function. The process involves repeatedly applying differentiation rules, such as the product rule, quotient rule, and chain rule, to find subsequent derivatives. This is essential in determining whether a given function satisfies a higher-order differential equation, as demonstrated in the original exercise where the fourth derivative of a cosine function was a key step in verifying the solution.

Chain Rule

The chain rule is a fundamental principle in calculus used when taking the derivative of a composite function. A composite function is created when one function is nested within another. In the given exercise, we encounter the composite function \( y = 3 \cos(2x) \) where the outer function is the cosine and the inner function is \( 2x \).

The chain rule states that to find the derivative of a composite function, you take the derivative of the outer function, leaving the inner function unchanged, and multiply it by the derivative of the inner function. This rule is repeatedly applied when taking successive derivatives of a function in higher-order differential equations, often leading to patterns that simplify the process. As seen, applying the chain rule properly is critical to accurately compute higher-order derivatives, ensuring that complex differential equations are solved correctly.

Cosine Function

The cosine function is one of the primary trigonometric functions, and understanding its properties is essential in solving differential equations. The function \( \cos(x) \) exhibits periodic behavior, meaning it repeats its values in regular intervals—this property is called 'periodicity.' Another important aspect of the cosine function is that it oscillates between -1 and 1.

In differential calculus, \( \cos(x) \) often appears within a composite function that requires the application of trigonometric identities and differentiation rules to solve. For instance, in our exercise, the cosine function's periodicity is reflected in the repetitive nature of its higher-order derivatives. Recognizing that after four derivatives, the cosine function essentially repeats its form, though potentially with altered amplitude and sign, is pivotal in solving related differential equations.

Differential Equation Solving

Solving a differential equation involves finding a function—or set of functions—that satisfies the equation when substituted along with its derivatives. An essential step in this process is verifying that the function in question aligns with the equation's order. A fourth-order differential equation, such as \( y^{(4)} - 16y = 0 \), requires the fourth derivative of the function to complete the solution process.

In solving these equations, the main goal is to determine if the original function and its derivatives fulfill the equation for all values of the variable, known as a general solution. The process, as illustrated in the exercise, involves taking the appropriate derivatives, substituting them back into the original equation, and simplifying to check for consistency. Mastery of solving differential equations is key to mathematical modeling in physics, engineering, and other scientific domains.