JEE Mains · Maths · STD 12 - 9. differential equations
Let \(f(x)=x-1\) and \(g(x)=e^x\) for \(x \in \mathbb{R}\). If \(\frac{d y}{d x}=\left(e^{-2 \sqrt{x}} g(f(f(x)))-\frac{y}{\sqrt{x}}\right), y(0)=0\), then \(y(1)\) is :-
- A \(\frac{1-e^2}{e^4}\)
- B \(\frac{2 e-1}{e^3}\)
- C \(\frac{e-1}{e^4}\)
- D \(\frac{1-e^3}{e^4}\)
Answer & Solution
Correct Answer
(C) \(\frac{e-1}{e^4}\)
Step-by-step Solution
Detailed explanation
\begin{aligned} & f(x)=x-1 \\ & f(f(x))=f(x)-1=x-1-1=x-2 \\ & g(f(f(x)))=e^{x-2} \\ & \therefore \frac{d y}{d x}=e^{-2 \sqrt{x}} \times e^{x-2}-\frac{1}{\sqrt{x}} y \\ & \frac{d y}{d x}+\frac{1}{\sqrt{x}} y=e^{x-2 \sqrt{x}-2} \text { which is L.D.E } \\ & \text { I.F. }=e^{\int…
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