Question

If \( y'(x) = 2y \), \( y(x) \ge 0 \) and \( y(0) = e^2 \), then \( y(x) = \)

• A. \( e^{x/2 + 2} \)
• B. \( e^{2x} \)
• C. \( e^{x/2} \)
• D. \( e^2 e^{2x} \)
• E. \( e^{2x} + 2 \)

Hint:

First solve general solution, then apply initial condition.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This is a first-order linear ordinary differential equation. Since it can be rearranged to have all `y` terms on one side and all `x` terms on the other, it is a separable equation. We will solve it by separating variables and integrating.
Step 2: Key Formula or Approach:
1. Rewrite the equation as \(y = 2 \frac{dy}{dx}\). 2. Separate the variables: move all `y` terms with `dy` and all `x` terms (in this case, just constants) with `dx`. 3. Integrate both sides of the equation. 4. Use the initial condition \(y(0) = e^2\) to solve for the constant of integration. 5. Write the particular solution for \(y(x)\).
Step 3: Detailed Explanation:
The given differential equation is: \[ y = 2 \frac{dy}{dx} \] Separate the variables by multiplying by `dx` and dividing by `y`: \[ \frac{1}{2} dx = \frac{1}{y} dy \] Now, integrate both sides: \[ \int \frac{1}{2} dx = \int \frac{1}{y} dy \] \[ \frac{x}{2} + C_1 = \ln|y| \] To solve for `y`, we exponentiate both sides: \[ e^{\frac{x}{2} + C_1} = |y| \] \[ |y| = e^{C_1} \cdot e^{x/2} \] Since we are given that \(y(x) \ge 0\), we can drop the absolute value. Let \(C = e^{C_1}\) be the new constant of integration. \[ y(x) = C e^{x/2} \] This is the general solution. Now, use the initial condition \(y(0) = e^2\) to find C. \[ y(0) = C e^{0/2} = C \cdot e^0 = C \cdot 1 = C \] So, \(C = e^2\). Substitute this value of C back into the general solution to get the particular solution: \[ y(x) = e^2 e^{x/2} \] This can also be written as \(y(x) = e^{x/2 + 2}\).
Step 4: Final Answer:
The solution is \(y(x) = e^2 e^{x/2}\).