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 in a subtraction form, always split the base using \( e^{a-b} = e^a/e^b \) to immediately identify the variable separable form.
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: If a differential equation can be written in the form \( f(y) dy = g(x) dx \), it can be solved by integrating both sides.
• Exponential properties: \( e^{a-b} = e^a \cdot e^{-b} = \frac{e^a}{e^b} \).
Step 1: Separate the variables \( x \) and \( y \)
The given differential equation is:
\[ \frac{dy}{dx} = e^{3x-y} \]
Using the law of exponents \( e^{A-B} = \frac{e^A}{e^B} \):
\[ \frac{dy}{dx} = \frac{e^{3x}}{e^y} \]
Rearrange the terms to put all \( y \) terms on the left and all \( x \) terms on the right:
\[ e^y \, dy = e^{3x} \, dx \]
Step 2: Integrate both sides of the equation
Apply the integral sign to both sides:
\[ \int e^y \, dy = \int e^{3x} \, dx \]
Recall the standard integral \( \int e^{ax} \, dx = \frac{e^{ax}}{a} + C \):
\[ e^y = \frac{e^{3x}}{3} + C_1 \]
Step 3: Simplify to match the given options
Multiply the entire equation by 3 to eliminate the fraction:
\[ 3e^y = e^{3x} + 3C_1 \]
Let \( 3C_1 = C \) (where \( C \) is a new arbitrary constant):
\[ 3e^y = e^{3x} + C \]
• Variable Separable Method: If a differential equation can be written in the form \( f(y) dy = g(x) dx \), it can be solved by integrating both sides.
• Exponential properties: \( e^{a-b} = e^a \cdot e^{-b} = \frac{e^a}{e^b} \).
Step 1: Separate the variables \( x \) and \( y \)
The given differential equation is:
\[ \frac{dy}{dx} = e^{3x-y} \]
Using the law of exponents \( e^{A-B} = \frac{e^A}{e^B} \):
\[ \frac{dy}{dx} = \frac{e^{3x}}{e^y} \]
Rearrange the terms to put all \( y \) terms on the left and all \( x \) terms on the right:
\[ e^y \, dy = e^{3x} \, dx \]
Step 2: Integrate both sides of the equation
Apply the integral sign to both sides:
\[ \int e^y \, dy = \int e^{3x} \, dx \]
Recall the standard integral \( \int e^{ax} \, dx = \frac{e^{ax}}{a} + C \):
\[ e^y = \frac{e^{3x}}{3} + C_1 \]
Step 3: Simplify to match the given options
Multiply the entire equation by 3 to eliminate the fraction:
\[ 3e^y = e^{3x} + 3C_1 \]
Let \( 3C_1 = C \) (where \( C \) is a new arbitrary constant):
\[ 3e^y = e^{3x} + C \]
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