Question:
If $y=\left(\tan ^{-1} x\right)^{2}$, show that $\left(x^{2}+1\right)^{2} y_{2}+2 x\left(x^{2}+1\right) y_{1}=2$
Solution:
The given relationship is $y=\left(\tan ^{-1} x\right)^{2}$
Then,
$y_{1}=2 \tan ^{-1} x \frac{d}{d x}\left(\tan ^{-1} x\right)$
$\Rightarrow y_{1}=2 \tan ^{-1} x \cdot \frac{1}{1+x^{2}}$
$\Rightarrow\left(1+x^{2}\right) y_{1}=2 \tan ^{-1} x$
Again differentiating with respect to $x$ on both the sides, we obtain
$\left(1+x^{2}\right) y_{2}+2 x y_{1}=2\left(\frac{1}{1+x^{2}}\right)$
$\Rightarrow\left(1+x^{2}\right)^{2} y_{2}+2 x\left(1+x^{2}\right) y_{1}=2$
Hence, proved.
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- 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
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- Atoms
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- Chemical Effects of Electric Current,
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