JEE Mains · Maths · STD 12 - 9. differential equations
If the solution curve \(f(x, y)=0\) of the differential equation \(\left(1+\log _e x\right) \frac{d x}{d y}-x \log _e x=e^y, x > 0\), passes through the points \((1,0)\) and \((\alpha, 2)\) then \(\alpha^\alpha\) is equal to
- A \(e ^{2 e ^{\sqrt{2}}}\)
- B \(e ^{\sqrt{2} e^2}\)
- C \(e ^{ e ^2}\)
- D \(e^{2 e^2}\)
Answer & Solution
Correct Answer
(D) \(e^{2 e^2}\)
Step-by-step Solution
Detailed explanation
\((1+\ln x) \frac{d x}{d y}-x \ln x=e^y\) Let \(x \ln x = t\) \((1+\ln x) \frac{d x}{d y}=\frac{d t}{d y}\) \(\frac{d t}{d y}-t=e^y\) \(\text { If }=e^{\int-d y}=e^{-y}\) \(t e^{-y}=\int e^y e^{-y} d y+c\) \(t e^{-y}=y+c\) \(x \ln x e^{-y}=y+c\) \(x \ln x=y e^y+c e^y\)…
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