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

Detailed Solution:

Step 1: Apply the natural logarithm to both sides

To eliminate the exponential function on the left side, we take the natural logarithm (ln) of both sides:

ln (e^½(dy/dx)) = ln (3^x)

Step 2: Use logarithmic identities to simplify both sides

• Using the identity ln(e^a) = a, the left side becomes ½(dy/dx)
• Using the identity ln(a^b) = b × ln(a), the right side becomes x × ln(3)

So we now have:

½(dy/dx) = x × ln(3)

Step 3: Multiply both sides by 2 to isolate dy/dx

dy/dx = 2x × ln(3)

Step 4: Integrate both sides with respect to their respective variables

We now integrate both sides:

∫ dy = ∫ 2x × ln(3) dx

Step 5: Pull out constants and integrate

Since ln(3) is a constant, we can factor it out:

y = 2ln(3) × ∫ x dx

Now integrate x:

∫ x dx = x² / 2

So,

y = 2ln(3) × (x² / 2) + C₁

Step 6: Simplify the expression

The 2 and ½ cancel out:

y = ln(3) × x² + C₁

Step 7: Generalize the constant

Since C₁ is just a constant of integration, we can rename it simply as C for simplicity:

Final Answer: y = x² log 3 + C

Therefore, the correct answer is: (B) y = x² log 3 + C