Let $y=y(x)$ be the solution of the differential equation, $\frac{d y}{d x}+y \tan x=2 x+x^{2} \tan x$
$x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, such that $y(0)=1$. Then :
$\mathrm{y}^{\prime}\left(\frac{\pi}{4}\right)+\mathrm{y}^{\prime}\left(\frac{-\pi}{4}\right)=-\sqrt{2}$
$y^{\prime}\left(\frac{\pi}{4}\right)-y^{\prime}\left(\frac{-\pi}{4}\right)=\pi-\sqrt{2}$
$y\left(\frac{\pi}{4}\right)-y\left(-\frac{\pi}{4}\right)=\sqrt{2}$
$\mathrm{y}\left(\frac{\pi}{4}\right)+\mathrm{y}\left(-\frac{\pi}{4}\right)=\frac{\pi^{2}}{2}+2$
Correct Option: , 2
$\frac{d y}{d x}+y(\tan x)=2 x+x^{2} \tan x$
I.F $=e^{\int \tan x d x}=e^{\ln \cdot \sec x}=\sec x$
$\therefore y \cdot \sec x=\int\left(2 x+x^{2} \tan x\right) \sec x \cdot d x$
$=\int 2 x \sec x d x+\int x^{2}(\sec x \cdot \tan x) d x$
$y \sec x=x^{2} \sec x+\lambda$
$\Rightarrow y=x^{2}+\lambda \cos x$
$y(0)=0+\lambda=1 \quad \Rightarrow \lambda=1$
$y=x^{2}+\cos x$
$y\left(\frac{\pi}{4}\right)=\frac{\pi^{2}}{16}+\frac{1}{\sqrt{2}}$
$y\left(-\frac{\pi}{4}\right)=\frac{\pi^{2}}{16}+\frac{1}{\sqrt{2}}$
$y^{\prime}(x)=2 x-\sin x$
$y^{\prime}\left(\frac{\pi}{4}\right)=\frac{\pi}{2}-\frac{1}{\sqrt{2}}$
$y^{\prime}\left(\frac{-\pi}{4}\right)=\frac{-\pi}{2}+\frac{1}{\sqrt{2}}$
$y^{\prime}\left(\frac{\pi}{4}\right)-y^{\prime}\left(\frac{-\pi}{4}\right)=\pi-\sqrt{2}$
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All Study Material
- JEE Main
- Exam Pattern
- Previous Year Papers
- PYQ Chapterwise
- Physics
- Kinematics 1D
- Kinemetics 2D
- Friction
- Work, Power, Energy
- Centre of Mass and Collision
- Rotational Dynamics
- Gravitation
- Calorimetry
- Elasticity
- Thermal Expansion
- Heat Transfer
- Kinetic Theory of Gases
- Thermodynamics
- Simple Harmonic Motion
- Wave on String
- Sound waves
- Fluid Mechanics
- Electrostatics
- Current Electricity
- Capacitor
- Magnetism and Matter
- Electromagnetic Induction
- Atomic Structure
- Dual Nature of Matter
- Nuclear Physics
- Radioactivity
- Semiconductors
- Communication System
- Error in Measurement & instruments
- Alternating Current
- Electromagnetic Waves
- Wave Optics
- X-Rays
- All Subjects
- Physics
- Motion in a Plane
- Law of Motion
- Work, Energy and Power
- Systems of Particles and Rotational Motion
- Gravitation
- Mechanical Properties of Solids
- Mechanical Properties of Fluids
- Thermal Properties of matter
- Thermodynamics
- Kinetic Theory
- Oscillations
- Waves
- Electric Charge and Fields
- Electrostatic Potential and Capacitance
- Current Electricity
- Thermoelectric Effects of Electric Current
- Heating Effects of Electric Current
- Moving Charges and Magnetism
- Magnetism and Matter
- Electromagnetic Induction
- Alternating Current
- Electromagnetic Wave
- Ray Optics and Optical Instruments
- Wave Optics
- Dual Nature of Radiation and Matter
- Atoms
- Nuclei
- Semiconductor Electronics: Materials, Devices and Simple Circuits.
- Chemical Effects of Electric Current,