JEE Mains · Maths · STD 12 - 9. differential equations
If \(y=y(x)\) is the solution of the differential equation \(\frac{d y}{d x}+2 y=\sin (2 x), y(0)=\frac{3}{4}\), then \(\mathrm{y}\left(\frac{\pi}{8}\right)\) is equal to :
- A \(\mathrm{e}^{-\pi / 8}\)
- B \(\mathrm{e}^{-\pi / 4}\)
- C \(e^{\pi / 4}\)
- D \(e^{\pi / 8}\)
Answer & Solution
Correct Answer
(B) \(\mathrm{e}^{-\pi / 4}\)
Step-by-step Solution
Detailed explanation
\( \frac{d y}{d x}+2 y=\sin 2 x, y(0)=\frac{3}{4} \) \( \text { I.F }=e^{\int 2 d x}=e^{2 x} \) \( y . e^{2 x}=\int e^{2 x} \sin 2 x d x \) \( y . e^{2 x}=\frac{e^{2 x}(2 \sin 2 x-2 \cos 2 x)}{4+4}+C \) \( x=0, y=\frac{3}{4} \Rightarrow \frac{3}{4} \cdot 1=\frac{1(0-2)}{8}+C \)…
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 \(y=y(x)\) be the solution of the differential equation \(x d y=\left(y+x^{3} \cos x\right) d x\) with \(y(\pi)=0\) then \(y\left(\frac{\pi}{2}\right)\) is equal to :JEE Mains 2021 Hard
- Consider the function \(\mathrm{f}:(0,2) \rightarrow \mathrm{R}\) defined by \(f(x)=\frac{x}{2}+\frac{2}{x}\) and the function\( g(x)\) defined by \(g(x)=\left\{\begin{array}{cc}\min \{f(t)\}, & 0 < t \leq x \text { and } 0 < x \leq 1 \\ \frac{3}{2}+x, & 1 < x< 2\end{array}\right.\). ThenJEE Mains 2024 Hard
- The function \(f ( x )= xe x ^{ x (1- x )}, x \in R\), isJEE Mains 2022 Medium
- If the mean of the frequency distribution
is \(28\) , then its variance is \(........\).Class: \(0-10\) \(10-20\) \(20-30\) \(30-40\) \(40-50\) Frequency \(2\) \(3\) \(x\) \(5\) \(4\) JEE Mains 2023 Hard - If \(m\) arithmetic means \(( A . Ms )\) and three geometric means \((G.Ms)\) are inserted between \(3\) and \(243\) such that \(4^{\text {th }}\) \(A.M.\) is equal to \(2^{\text {nd }}\) \(G.M.\), then \(m\) is equal toJEE Mains 2020 Hard
- If \( y=y(x) \) satisfies the differential equation \( 16(\sqrt{x+9\sqrt{x}})(4+\sqrt{9+\sqrt{x}})cos~y~dy=(1+2~sin~y)dx, x>0 \) and \( y(256)=\frac{\pi}{2}, y(49)=\alpha \) then \( 2~sin~\alpha \) is equal to:JEE Mains 2026 Easy
More PYQs from JEE Mains
- Let the normals at all the points on a given curve pass through a fixed point \((a, b) .\) If the curve passes through \((3,-3)\) and \((4,-2 \sqrt{2}),\) and given that \(a-2 \sqrt{2} b=3,\) then \(\left(a^{2}+b^{2}+a b\right)\) is equal to ..... .JEE Mains 2021 Hard
- The product of all the rational roots of the equation \(\left(x^2-9 x+11\right)^2-(x-4)(x-5)=3\), is equal toJEE Mains 2025 Medium
- If the function \(f(x)=\left(\frac{1}{x}\right)^{2 x} ; x>0\) attains the maximum value at \(\mathrm{x}=\frac{1}{\mathrm{e}}\) then :JEE Mains 2024 Hard
- The coefficient of \(x^{7}\) in the expression \((1+x)^{10}+x(1+x)^{9}+x^{2}(1+x)^{8}+\ldots+x^{10}\) isJEE Mains 2020 Hard
- Let \(H : \frac{ x ^{2}}{ a ^{2}}-\frac{y^{2}}{ b ^{2}}=1\), a \(>0, b >0\), be a hyperbola such that the sum of lengths of the transverse and the conjugate axes is \(4(2 \sqrt{2}+\sqrt{14})\). If the eccentricity \(H\) is \(\frac{\sqrt{11}}{2}\), then value of \(a^{2}+b^{2}\) is equal toJEE Mains 2022 Hard
- Let \(S = \left\{ {\left( {x,y} \right) \in {R^2}:\frac{{{y^2}}}{{1 + r}} - \frac{{{x^2}}}{{1 - r}} = 1} \right\}\), where \(r \ne \pm 1\). Then \(S\) representsJEE Mains 2019 Hard