JEE Mains · Maths · STD 12 - 9. differential equations
If \(\frac{d y}{d x}=\frac{2^{x+y}-2^{x}}{2^{y}}, y(0)=1\), then \(y(1)\) is equal to :
- A \(\log _{2}(2+\mathrm{e})\)
- B \(\log _{2}(1+\mathrm{e})\)
- C \(\log _{2}(2 \mathrm{e})\)
- D \(\log _{2}\left(1+\mathrm{e}^{2}\right)\)
Answer & Solution
Correct Answer
(B) \(\log _{2}(1+\mathrm{e})\)
Step-by-step Solution
Detailed explanation
\(\frac{d y}{d x}=\frac{2^{x} 2^{y}-2^{x}}{2^{y}}\) \(2^{y} \frac{d y}{d x}=2^{x}\left(2^{y}-1\right)\) \(\int \frac{2^{y}}{2^{y}-1} d y=\int 2^{x} d x\) \(\frac{\ln \left(2^{y}-1\right)}{\ln 2}=\frac{2^{x}}{\ln 2}+C\)…
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