JEE Mains · Maths · STD 12 - 9. differential equations
Let \(x = x ( y )\) be the solution of the differential equation \(2( y +2) \log _e( y +2) dx +\left( x +4-2 \log _e( y +2)\right) dy =0\), \(y > -1\) with \(x\left(e^4-2\right)=1\). Then \(x\left(e^9-2\right)\) is equal to
- A \(\frac{4}{9}\)
- B \(\frac{10}{3}\)
- C \(\frac{32}{9}\)
- D \(3\)
Answer & Solution
Correct Answer
(C) \(\frac{32}{9}\)
Step-by-step Solution
Detailed explanation
\(2(y+2) \ln (y+2) d x+(x+4-2 \ln (y+2)) d y=0\) \(2 \ln (y+2)+(x+4-2 \ln (y+2)) \frac{1}{y+2} \cdot \frac{d y}{d x}=0\) \(\text { let, } \ln (y+2)=t\) \(\frac{1}{y+2} \cdot \frac{d y}{d x}=\frac{d t}{d x}\) \(2 t+(x+4-2 t) \cdot \frac{d t}{d x}=0\)…
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