Question

The general solution of differential equation \(e^{\frac {1}{2} (\frac {dy}{dx})}\) = 3^x is (where C is a constant of integration.)

• A. x = (log 3)y² + C
• B. y = x²log 3 + C
• C. y = xlog 3 + C
• D. y = 2xlog 3 + C

Answer 1

The Correct Option is B

\(e^{\frac {1}{2} (\frac {dy}{dx})}\) = 3^x
Taking the natural logarithm of both sides:
ln (\(e^{\frac {1}{2} (\frac {dy}{dx})}\)) = ln (3^x) 
\(\frac {1}{2} (\frac {dy}{dx})\) = x ln (3) 
\(\frac {dy}{dx}\) = 2x ln (3) 
Now, we can integrate both sides with respect to their respective variables: 
∫dy = ∫2x ln (3) dx 
y = 2ln (3) x ∫x dx 
y = 2ln(3) c \((\frac {x^2}{2})\) + C₁ 
y = ln(3) . x² + C₁
Since C₁ is an arbitrary constant, we can rewrite it as another constant C: 
y = x² log 3 + C 
Therefore, the correct answer is option (B) y = x² log 3 + C.