Question

Differential Equations In Exercises \(125-128\) , verify that the function satisfies the differential equation. $$ \text{Function} \quad \text{Differential Equation} $$ $$ y=2 \sin x+3 \quad y^{\prime \prime}+y=3 $$

Short answer

Yes, the function \(y=2 \sin x+3\) does satisfy the given differential equation \(y^{\prime \prime}+y=3\).

Step-by-step solution

Calculate the first derivative

Differentiating the function \(y=2 \sin x+3\) once with respect to \(x\), the first derivative \(y'\) is given as: \(y' = 2 \cos x\) using chain rule of differentiation.

Calculate the second derivative

Differentiating \(y'\) once more with respect to \(x\) gives the second derivative \(y''\) as: \(y'' = -2 \sin x\).

Substituting into the differential equation

Substitute the original function \(y\) and its second derivative \(y''\) into the original differential equation. This gives: \(-2 \sin x + 2 \sin x + 3 = 3\). The left side of the equation simplifies to \(3\), which is equal to the right side. This confirms that the given function satisfies the differential equation.

Key concepts

Verification of Solutions

To verify if a function is a solution to a differential equation, we check if the function, when substituted into the equation, makes a true statement. This is a crucial step in solving differential equations. The process includes:

• Calculating the necessary derivatives of the function.
• Replacing the function and its derivatives into the differential equation.
• Checking if both sides of the equation are equivalent after substitution.

For the given function, \[ y = 2 \sin x + 3, \]we need its second derivative and the function itself to substitute into \[ y'' + y = 3. \]After substitution, if both sides equal, the function is indeed a solution. This method confirms mathematical consistency and correctness of potential solutions.

Chain Rule

The chain rule is a fundamental calculus concept used to find the derivative of composite functions. When dealing with such functions, the chain rule helps differentiate each layer of the composition. For instance, if you have \( y = 2 \sin x + 3 \),the derivative involves applying the chain rule.

- The outer function is the constant multiple of the inner function, \( 2 \sin x \), resulting in a direct differentiation.- Inside, you differentiate \( \sin x \) which provides \( \cos x \) as its derivative.
Using this formula helps reduce errors and simplifies differentiation of complex expressions. By chaining derivatives, it allows for a step-by-step extraction of each function's derivative within a composite function.

Second Derivative

The second derivative represents the derivative of the first derivative. It gives us insight into the curvature or concavity of the function's graph. To find the second derivative, differentiate the expression for the first derivative again.

For our function \[ y = 2 \sin x + 3, \]the first derivative is \( y' = 2 \cos x \).Differentiating again using standard trigonometric derivatives, we find:

• \( y'' = -2 \sin x \).

This derivative is critical in verifying solutions to differential equations, providing a piece which, combined with the original function, allows us to solve differential equations like \( y'' + y = 3. \).

Function Substitution

Function substitution involves replacing a function and its derivatives into a given equation to verify if it holds. This method is used to test potential solutions for differential equations. Here's how it works:

• First, identify the given differential equation and the candidate function.
• Next, calculate all necessary derivatives of the function.
• Then, substitute the function and its derivatives into the differential equation.

For example, using the function \[ y = 2 \sin x + 3 \]and its second derivative \[ y'' = -2 \sin x, \]substitute back to verify:\[ y'' + y = -2 \sin x + 2 \sin x + 3 = 3. \]The equation simplifies correctly to 3, showing that the original function is indeed a valid solution.