JEE Mains · Maths · STD 12 - 9. differential equations
Let \( y=y(x) \) be a differentiable function in the interval \( (0, \infty) \) such that \( y(1)=2 \) and \( \lim_{t\rightarrow x}(\frac{t^{2}y(x)-x^{2}y(t)}{x-t})=3 \) for each \( x>0. \) Then \( 2y(2) \) is equal to
- A 18
- B 23
- C 27
- D 12
Answer & Solution
Correct Answer
(B) 23
Step-by-step Solution
Detailed explanation
\(\lim _{t \rightarrow x} \frac{2 t f(x)-x^2 f^{\prime}(t)}{-1}=3\) \(x^2 f^{\prime}(x)-2 x f(x)=3\) \(\frac{d y}{d x}-\frac{2 y}{x}=\frac{3}{x^2}\) \(\text { I.F. }= e ^{-\int \frac{2}{ x } dx }= e ^{-2 \log _{ e } x }=1 / x ^2\)…
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