Question

The general solution of differential equation $e^{\frac {1}{2} (\frac {dy}{dx})}$ = 3x 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

Answer: (B)

y = x2log 3 + C

Answer 2

$e^{\frac {1}{2} (\frac {dy}{dx})}$ = 3x
Taking the natural logarithm of both sides:
ln ($e^{\frac {1}{2} (\frac {dy}{dx})}$) = ln (3x)
$\frac {1}{2} (\frac {dy}{dx})$ = 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.