Substituting into the Differential Equation

Oct 13, 2025 by magicalceylontours.com 69 views

What Value of k Makes y a Solution to the Differential Equation?

In this article, we will explore the value of k that makes the given function y a solution to the differential equation 4y - y'' = 10e^(-3x). To do this, we will first find the second derivative of the function y, and then substitute it into the differential equation. We will then solve for k to determine the value that makes y a solution to the equation.

The given function is y = e^(2x) + ke^(-3x), where k is a constant. To find the second derivative of this function, we will first find the first derivative using the chain rule.

First Derivative

To find the first derivative of y, we will use the chain rule. The derivative of e^(2x) is 2e^(2x), and the derivative of ke^(-3x) is -3ke^(-3x). Therefore, the first derivative of y is:

y' = 2e^(2x) - 3ke^(-3x)

Second Derivative

To find the second derivative of y, we will differentiate the first derivative. Using the chain rule again, we get:

y'' = 4e^(2x) + 9ke^(-3x)

Substituting into the Differential Equation

Now that we have found the second derivative of y, we can substitute it into the differential equation 4y - y'' = 10e^(-3x). Substituting y and y'' into the equation, we get:

4(e^(2x) + ke^(-3x)) - (4e^(2x) + 9ke^(-3x)) = 10e^(-3x)

Simplifying the Equation

Simplifying the equation, we get:

4e^(2x) + 4ke^(-3x) - 4e^(2x) - 9ke^(-3x) = 10e^(-3x)

Combining like terms, we get:

-5ke^(-3x) = 10e^(-3x)

Solving for k

To solve for k, we can divide both sides of the equation by -5e^(-3x). This gives us:

k = -2

Therefore, the value of k that makes y a solution to the differential equation is -2.

In this article, we found the value of k that makes the given function y a solution to the differential equation 4y - y'' = 10e^(-3x). We first found the second derivative of y, and then substituted it into the differential equation. Solving for k, we found that k = -2.

The final answer is: $\boxed{-2}$
Q&A: What Value of k Makes y a Solution to the Differential Equation?

In our previous article, we explored the value of k that makes the given function y a solution to the differential equation 4y - y'' = 10e^(-3x). We found that k = -2. In this article, we will answer some frequently asked questions about the problem.

Q: What is the given function y?

A: The given function y is e^(2x) + ke^(-3x), where k is a constant.

Q: What is the differential equation?

A: The differential equation is 4y - y'' = 10e^(-3x).

Q: How do we find the second derivative of y?

A: To find the second derivative of y, we first find the first derivative using the chain rule. The derivative of e^(2x) is 2e^(2x), and the derivative of ke^(-3x) is -3ke^(-3x). Therefore, the first derivative of y is 2e^(2x) - 3ke^(-3x). We then differentiate the first derivative to find the second derivative.

Q: What is the second derivative of y?

A: The second derivative of y is 4e^(2x) + 9ke^(-3x).

Q: How do we substitute the second derivative into the differential equation?

A: We substitute the second derivative into the differential equation 4y - y'' = 10e^(-3x). This gives us 4(e^(2x) + ke^(-3x)) - (4e^(2x) + 9ke^(-3x)) = 10e^(-3x).

Q: How do we simplify the equation?

A: We simplify the equation by combining like terms. This gives us -5ke^(-3x) = 10e^(-3x).

Q: How do we solve for k?

A: We solve for k by dividing both sides of the equation by -5e^(-3x). This gives us k = -2.

Q: What is the final answer?

A: The final answer is k = -2.

Q: What if the differential equation is 4y - y'' = 10e^(3x)? A: In this case, we would find the second derivative of y and substitute it into the differential equation. We would then solve for k.
Q: What if the function y is y = e^(2x) + ke^(3x)? A: In this case, we would find the second derivative of y and substitute it into the differential equation. We would then solve for k.
Q: How do we know if the function y is a solution to the differential equation? A: We know if the function y is a solution to the differential equation by substituting the function and its derivatives into the differential equation and solving for k.

In this article, we answered some frequently asked questions about the problem of finding the value of k that makes the given function y a solution to the differential equation 4y - y'' = 10e^(-3x). We hope that this article has been helpful in understanding the problem and its solution.