JEE Mains · Maths · STD 12 - 9. differential equations
Which of the following is true for \(y ( x )\) that satisfies the differential equation \(\frac{d y}{d x}=x y-1+x-y ; y(0)=0\)
- A \(y (1)= e ^{-\frac{1}{2}}-1\)
- B \(y(1)=e^{\frac{1}{2}}-e^{-\frac{1}{2}}\)
- C \(y(1)=1\)
- D \(y(1)=e^{\frac{1}{2}}-1\)
Answer & Solution
Correct Answer
(A) \(y (1)= e ^{-\frac{1}{2}}-1\)
Step-by-step Solution
Detailed explanation
\(\frac{d y}{d x}=(1+y)(x-1)\) \(\frac{d y}{(y+1)}=(x-1) d x\) Integrate \(\ln (y+1)=\frac{x^{2}}{2}-x+c\) \((0,0) \Rightarrow c=0 \Rightarrow y=e^{\left(\frac{x^{2}}{2}-x\right)}-1\)
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