Let x = x(y) be the solution of the differential equation 2(y+2) loge (y+2)dx+(x+4-2loge(y+2))dy = 0, y >-1 with x (e4-2)=1. Then x(e9-2) is equal to
- 3
- 10/3
- 4/9
- 32/9
The Correct Option is D
Solution and Explanation
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. $\frac{32}{9}$
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