Question

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

Short answer

The function \(y = e^{-2x}\) is not a solution to the differential equation \(y^{(4)} - 16y = 0\) given that upon substitution, the equation is not equated to zero.

Step-by-step solution

Differentiating Function Y

Differentiate the function \(y = e^{-2x}\) four times to find \(y{'}'\), \(y{''}\), \(y^{(3)}\) and \(y^{(4)}\). This can be done using the chain rule of differentiation.

Substituting y-values

Substitute the calculated values of y and \(y^{(4)}\) into the given differential equation \(y^{(4)} - 16y = 0\) and simplify.

Verifying the Equation

If the equation is satisfied, that is equals zero, then \(y = e^{-2x}\) is a solution to the differential equation \(y^{(4)} - 16y = 0\). Otherwise, if the equation is not satisfied, the function is not a solution.

Key concepts

Chain Rule

The chain rule is an essential tool in calculus, particularly when working with composite functions. It allows you to differentiate functions where one function is nested inside another. If you have a composite function of the form \( u(g(x)) \), the chain rule states that the derivative is \( u'(g(x)) \cdot g'(x) \). You essentially differentiate the outer function, keeping the inner function unchanged, and then multiply by the derivative of the inner function.

• Example: For \( y = e^{-2x} \), identify two parts: the outside function is \( e^u \) where \( u = -2x \), and the derivative of \( e^u \) is \( e^u \).
• The inside function \( u = -2x \) has a derivative of \(-2\).
• Combining these results, the derivative \( y' \) becomes \(-2e^{-2x} \).

By using this method repeatedly, you can find higher-order derivatives, such as the fourth derivative, which is necessary to solve the given differential equation.

Differentiation

Differentiation is the process of finding the derivative of a function. The derivative represents how a function changes as its input changes. To differentiate a function like \( y = e^{-2x} \), you use the chain rule, as it involves a composite function structure.

• First Derivative: Using the chain rule, the first derivative \( y' = -2e^{-2x} \).
• Second Derivative: Differentiate \(-2e^{-2x} \) to get the second derivative \( y'' = 4e^{-2x} \).
• Third Derivative: Differentiate \( 4e^{-2x} \) to find \( y^{(3)} = -8e^{-2x} \).
• Fourth Derivative: Finally, differentiate \( -8e^{-2x} \) to obtain \( y^{(4)} = 16e^{-2x} \).

This process shows the systematic way of finding successive derivatives, which is necessary for solving the differential equation \( y^{(4)} - 16y = 0 \).

Fourth Derivative

The fourth derivative of a function gives insights into the behavior of its underlying changes and oscillations. For the function \( y = e^{-2x} \), the fourth derivative \( y^{(4)} = 16e^{-2x} \) shows that the form of the function doesn't change; it's a scalar multiple of the original function distributed through each differentiation step.

• Purpose in Differential Equations: The fourth derivative is crucial in solving higher-order differential equations. Here, substituting \( y^{(4)} \) and \( y \) into \( y^{(4)} - 16y = 0 \) determines if \( e^{-2x} \) satisfies this equation.
• Verification: Substitute \( y^{(4)} = 16e^{-2x} \) and the original \( y = e^{-2x} \) into the differential equation.
• Find that \( 16e^{-2x} - 16e^{-2x} = 0 \), resulting in a true statement, confirming \( y = e^{-2x} \) as a solution.

Understanding the fourth derivative is key in verifying and forming correct higher-order solutions.