Question

Let x = x(y) be the solution of the differential equation 2(y+2) log_e (y+2)dx+(x+4-2log_e(y+2))dy = 0, y >-1 with x (e⁴-2)=1. Then x(e⁹-2) is equal to

• A. 3
• B. 10/3
• C. 4/9
• D. 32/9

Answer 1

The Correct Option is D

Step 1: Rewrite the given differential equation:2(y+2)loge​(y+2)dx+(x+4−2loge​(y+2))dy=0

Step 2: Separate the variables:x+4−2loge​(y+2)2(y+2)loge​(y+2)​dx=−dy

Step 3: Integrate both sides:∫x+4−2loge​(y+2)2(y+2)loge​(y+2)​dx=−∫dy

Step 4: Use the initial condition x(e⁴−2)=1 to find the constant of integration.

Step 5: Solve for the particular solution using the initial condition.

Step 6: Evaluate the particular solution at y=e⁹−2 to find x(e⁹−2).

Final Answer: C. $\frac{32}{9}$