JEE Mains · Maths · STD 12 - 9. differential equations
Let \( y=y(x) \) be the solution curve of the differential equation \( (1+x^{2})dy+(y-\tan^{-1}x)dx=0, \) \( y(0)=1 \). Then the value of \( y(1) \) is:
- A \( \frac{2}{e^{\frac{\pi}{4}}}+\frac{\pi}{4}-1 \)
- B \( \frac{2}{e^{\frac{\pi}{4}}}-\frac{\pi}{4}-1 \)
- C \( \frac{4}{e^{\frac{\pi}{4}}}+\frac{\pi}{2}-1 \)
- D \( \frac{4}{e^{\frac{\pi}{4}}}-\frac{\pi}{2}-1 \)
Answer & Solution
Correct Answer
(A) \( \frac{2}{e^{\frac{\pi}{4}}}+\frac{\pi}{4}-1 \)
Step-by-step Solution
Detailed explanation
\( \frac{dy}{dx}+\frac{y}{x^{2}+1}=\frac{\tan^{-1}x}{x^{2}+1} \) \( I.f.=e^{\tan^{-1}x} \) \( y\times e^{\tan^{-1}x}=\int e^{\tan^{-1}x}.\frac{\tan^{-1}x}{1+x^{2}}dx \) \( y\times e^{\tan^{-1}x}=\tan^{-1}x(e^{\tan^{-1}x})-e^{\tan^{-1}x}+c \) \( y(0)=1\Rightarrow c=2 \)…
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- Let \(A=\{1,3,4,6,9\}\) and \(B=\{2,4,5,8,10\}\). Let \(R\) be a relation defined on \(A \times B\) such that \(R =\) \(\left\{\left(\left(a_1, b_1\right),\left(a_2, b_2\right)\right): a_1 \leq b_2\right.\) and \(\left.b_1 \leq a_2\right\}\). Then the number of elements in the set \(R\) isJEE Mains 2023 Hard
- If \(\alpha\) and \(\beta\) are the roots of the equation \(2 x (2 x +1)=1,\) then \(\beta\) is equal toJEE Mains 2020 Medium
- The square of the distance of the point of intersection of the line \(\frac{x-1}{2}=\frac{y-2}{3}=\frac{z+1}{6}\) and the plane \(2 x-y+z=6\) from the point \((-1,-1,2)\) is .... .JEE Mains 2021 Medium
- The value of \(\left(\frac{-1+i \sqrt{3}}{1-i}\right)^{30}\) isJEE Mains 2020 Hard
- If \(1+\left(2+{ }^{49} C _{1}+{ }^{49} C _{2}+\ldots .+{ }^{49} C _{49}\right)\left({ }^{50} C _{2}+{ }^{50} C _{4}+\right.\) \(\ldots . .+{ }^{50} C _{ so }\) ) is equal to \(2^{ n } . m\), where \(m\) is odd, then \(n\) \(+m\) is equal to.JEE Mains 2022 Hard
- Four fair dice are thrown independently \(27\) times. Then the expected number of times, at least two dice show up a three or a five, isJEE Mains 2020 Hard
More PYQs from JEE Mains
- The system of equations \(kx + y + z =1\) \(x + ky + z = k\) and \(x + y + zk = k ^{2}\) has no solution if \(k\) is equal toJEE Mains 2021 Medium
- Let \(I(x)=\int \frac{x^2\left(x \sec ^2 x+\tan x\right)}{(x \tan x+1)^2} d x\). If \(I(0)=0\) the \(I\) \(\left(\frac{\pi}{4}\right)\) is equal toJEE Mains 2023 Hard
- Let \(A =\left(\begin{array}{cc} m & n \\ p & q \end{array}\right), d =| A | \neq 0| A - d (\operatorname{Adj} A )|=0\). ThenJEE Mains 2023 Hard
- The total number of functions,\(f:\{1,2,3,4\} \cdot\{1,2,3,4,5,6\}\) such that \(f (1)+ f (2)= f (3)\), is equal to .JEE Mains 2022 Hard
- For a differentiable function \(\mathrm{f}: I R \rightarrow I R\), suppose \(f^{\prime}(\mathrm{x})=3 f(\mathrm{x})+\alpha\), where \(\alpha \in \operatorname{IR}, f(0)=1\) and \(\lim _{x \rightarrow-\infty} f(x)=7\). Then \(9 \mathrm{f}\left(-\log _{\mathrm{e}} 3\right)\) is equal to ............JEE Mains 2024 Hard
- Let a be an integer such that \(\lim \limits_{x \rightarrow 7} \frac{18-[1-x]}{[x-3 a]}\) exists, where \([ t ]\) is greatest integer \(\leq t\). Then a is equal toJEE Mains 2022 Hard