Question:
If \( y'(x) = 2y \), \( y(x) \ge 0 \) and \( y(0) = e^2 \), then \( y(x) = \)
If \( y'(x) = 2y \), \( y(x) \ge 0 \) and \( y(0) = e^2 \), then \( y(x) = \)
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First solve general solution, then apply initial condition.
Updated On: Apr 21, 2026
- \( e^{x/2 + 2} \)
- \( e^{2x} \)
- \( e^{x/2} \)
- \( e^2 e^{2x} \)
- \( e^{2x} + 2 \)
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The Correct Option is D
Solution and Explanation
Concept:
Solve differential equation.
Step 1: Separate variables.
\[ \frac{dy}{y} = 2dx \]
Step 2: Integrate.
\[ \ln y = 2x + C \Rightarrow y = Ce^{2x} \]
Step 3: Apply condition.
\[ y(0)=e^2 \Rightarrow C = e^2 \] \[ y = e^2 e^{2x} \]
Step 1: Separate variables.
\[ \frac{dy}{y} = 2dx \]
Step 2: Integrate.
\[ \ln y = 2x + C \Rightarrow y = Ce^{2x} \]
Step 3: Apply condition.
\[ y(0)=e^2 \Rightarrow C = e^2 \] \[ y = e^2 e^{2x} \]
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