JEE Mains · Maths · STD 12 - 9. differential equations
Let \(y = y ( x )\) be the solution of the differential equation \(x d y-y d x=\sqrt{\left(x^{2}-y^{2}\right)} d x, x \geq 1\), with \(y (1)=0 .\) If the area bounded by the line \(x =1, x = e ^{\pi}, y =0\) and \(y = y ( x )\) is \(\alpha e ^{2 \pi}+\beta\) then the value of \(10(\alpha+\beta)\) is equal to ....... .
- A \(6\)
- B \(2\)
- C \(4\)
- D \(0\)
Answer & Solution
Correct Answer
(C) \(4\)
Step-by-step Solution
Detailed explanation
\(x d y-y d x=\sqrt{x^{2}-y^{2}} d x\) \(\Rightarrow \frac{x d y-y d x}{x^{2}}=\frac{1}{x} \sqrt{1-\frac{y^{2}}{x^{2}}} d x\) \(\Rightarrow \int \frac{d\left(\frac{y}{x}\right)}{\sqrt{1-\left(\frac{y}{x}\right)^{2}}}=\int \frac{d x}{x}\)…
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