Question:
The general solution for the differential equation \( \frac{dy}{dx} = e^{3x-y} \) is :
The general solution for the differential equation \( \frac{dy}{dx} = e^{3x-y} \) is :
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When variables are in the exponent, separating them usually transforms the problem into simple exponential integrals.
Constants can be absorbed into a single \( C \) at the end.
Constants can be absorbed into a single \( C \) at the end.
Updated On: Sep 10, 2026
- \( 3e^y = e^{3x} + C \)
- \( \log(3x - y) = C \)
- \( e^{3x-y} = C \)
- \( -e^y + 3e^{3x} = C \)
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The Correct Option is A
Solution and Explanation
Concept:
• Variable Separable Method: Group all \( y \) terms with \( dy \) and \( x \) terms with \( dx \).
• Use exponent laws: \( e^{a-b} = \frac{e^a}{e^b} \).
Step 1: Separate the variables
Given: \( \frac{dy}{dx} = e^{3x} \cdot e^{-y} \) \[ \frac{dy}{dx} = \frac{e^{3x}}{e^y} \] Multiplying both sides by \( e^y \, dx \): \[ e^y \, dy = e^{3x} \, dx \]
Step 2: Integrate both sides
\[ \int e^y \, dy = \int e^{3x} \, dx \] Applying standard integration formulas: \[ e^y = \frac{e^{3x}}{3} + C_1 \] where \( C_1 \) is the constant of integration.
Step 3: Simplify the expression to match options
Multiply the entire equation by \( 3 \): \[ 3e^y = e^{3x} + 3C_1 \] Let \( 3C_1 = C \) (a new constant): \[ 3e^y = e^{3x} + C \] This matches option (A).
• Variable Separable Method: Group all \( y \) terms with \( dy \) and \( x \) terms with \( dx \).
• Use exponent laws: \( e^{a-b} = \frac{e^a}{e^b} \).
Step 1: Separate the variables
Given: \( \frac{dy}{dx} = e^{3x} \cdot e^{-y} \) \[ \frac{dy}{dx} = \frac{e^{3x}}{e^y} \] Multiplying both sides by \( e^y \, dx \): \[ e^y \, dy = e^{3x} \, dx \]
Step 2: Integrate both sides
\[ \int e^y \, dy = \int e^{3x} \, dx \] Applying standard integration formulas: \[ e^y = \frac{e^{3x}}{3} + C_1 \] where \( C_1 \) is the constant of integration.
Step 3: Simplify the expression to match options
Multiply the entire equation by \( 3 \): \[ 3e^y = e^{3x} + 3C_1 \] Let \( 3C_1 = C \) (a new constant): \[ 3e^y = e^{3x} + C \] This matches option (A).
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