Question

If y = 3e2x + 2e3x, then $\frac{d^2y}{dx^2} + 6y$ is equal to

• A. $\frac{dy}{dx}$
• B. $5\frac{dy}{dx}$
• C. $6\frac{dy}{dx}$
• D. $30\frac{dy}{dx}$

Hint:

For problems of this type, which relate a function to its derivatives, recognize the structure of a linear homogeneous differential equation. The function \(y = C_1e^{r_1x} + C_2e^{r_2x}\) is the solution to \(y'' - (r_1+r_2)y' + r_1r_2y = 0\). Here, \(r_1=2\) and \(r_2=3\), so it's a solution to \(y'' - 5y' + 6y = 0\). From this, we can directly see that \(y'' + 6y = 5y'\).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This question involves finding the first and second derivatives of a function and then substituting them into a given expression to simplify it.
Step 2: Key Formula or Approach:
The key differentiation rule needed is for the exponential function:
\[ \frac{d}{dx}(e^{ax}) = a e^{ax} \] Step 3: Detailed Explanation:
The given function is \(y = 3e^{2x} + 2e^{3x}\).
First, let's find the first derivative, \(\frac{dy}{dx}\).
\[ \frac{dy}{dx} = \frac{d}{dx}(3e^{2x} + 2e^{3x}) = 3(2e^{2x}) + 2(3e^{3x}) \] \[ \frac{dy}{dx} = 6e^{2x} + 6e^{3x} \] Next, let's find the second derivative, \(\frac{d^2y}{dx^2}\).
\[ \frac{d^2y}{dx^2} = \frac{d}{dx}(6e^{2x} + 6e^{3x}) = 6(2e^{2x}) + 6(3e^{3x}) \] \[ \frac{d^2y}{dx^2} = 12e^{2x} + 18e^{3x} \] Now, we need to evaluate the expression \(\frac{d^2y}{dx^2} + 6y\).
Substitute the expressions for \(\frac{d^2y}{dx^2}\) and \(y\):
\[ \frac{d^2y}{dx^2} + 6y = (12e^{2x} + 18e^{3x}) + 6(3e^{2x} + 2e^{3x}) \] \[ = 12e^{2x} + 18e^{3x} + 18e^{2x} + 12e^{3x} \] Group the like terms:
\[ = (12 + 18)e^{2x} + (18 + 12)e^{3x} \] \[ = 30e^{2x} + 30e^{3x} \] Now, we compare this result with the options, which are in terms of \(\frac{dy}{dx}\).
Let's factor out 5 from our result:
\[ 30e^{2x} + 30e^{3x} = 5(6e^{2x} + 6e^{3x}) \] We recognize that the term in the parenthesis is exactly \(\frac{dy}{dx}\).
\[ \frac{d^2y}{dx^2} + 6y = 5\left(\frac{dy}{dx}\right) \] Step 4: Final Answer:
The expression \(\frac{d^2y}{dx^2} + 6y\) is equal to $5\frac{dy{dx}$}.