JEE Mains · Maths · STD 12 - 9. differential equations
Let \(y=y(x)\) be the solution of the differential equation \(\frac{d y}{d x}=(y+1)\left((y+1) e^{x^{2} / 2}-x\right), y(2)=0\) then \(y'(1)\) equal to . . . .
- A \(\frac{- e ^{3 / 2}}{\left( e ^{2}+1\right)^{2}}\)
- B \(-\frac{2 e^{2}}{\left(1+e^{2}\right)^{2}}\)
- C \(\frac{e^{5 / 2}}{\left(1+e^{2}\right)^{2}}\)
- D \(\frac{5 e ^{1 / 2}}{\left( e ^{2}+1\right)^{2}}\)
Answer & Solution
Correct Answer
(A) \(\frac{- e ^{3 / 2}}{\left( e ^{2}+1\right)^{2}}\)
Step-by-step Solution
Detailed explanation
Let \(y +1= Y\) \(\therefore \frac{ d Y }{ dx }= Y ^{2} e ^{\frac{ x ^{2}}{2}}- x Y\) Put \(-\frac{1}{ Y }= k\) \(\Rightarrow \frac{ d k }{ dx }+ k (- x )= e ^{\frac{ x ^{2}}{2}}\) \(I.F.= e ^{-\frac{ x ^{2}}{2}}\) \(\therefore k =( x + c ) e ^{ x ^{2} / 2}\) Put…
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