$\cos ^{2} x \frac{d y}{d x}+y=\tan x\left(0 \leq x<\frac{\pi}{2}\right)$
The given differential equation is:
$\cos ^{2} x \frac{d y}{d x}+y=\tan x$
$\Rightarrow \frac{d y}{d x}+\sec ^{2} x \cdot y=\sec ^{2} x \tan x$
This equation is in the form of:
$\frac{d y}{d x}+p y=Q$ (where $p=\sec ^{2} x$ and $Q=\sec ^{2} x \tan x$ )
Now, I.F $=e^{\int p d x}=e^{\int \sec ^{2} x d x}=e^{\tan x}$.
The general solution of the given differential equation is given by the relation,
$y(\mathrm{I.F} .)=\int(\mathrm{Q} \times \mathrm{I.F} .) d x+\mathrm{C}$
$\Rightarrow y \cdot e^{\tan x}=\int e^{\tan x} \cdot \sec ^{2} x \tan x d x+\mathrm{C}$ ...(1)
Let $\tan x=t .$
$\Rightarrow \frac{d}{d x}(\tan x)=\frac{d t}{d x}$
$\Rightarrow \sec ^{2} x=\frac{d t}{d x}$
$\Rightarrow \sec ^{2} x d x=d t$
Therefore, equation (1) becomes:
$y \cdot e^{\tan x}=\int\left(e^{t} \cdot t\right) d t+\mathrm{C}$
$\Rightarrow y \cdot e^{\tan x}=\int\left(t \cdot e^{t}\right) d t+\mathrm{C}$
$\Rightarrow y \cdot e^{\tan x}=t \cdot \int e^{t} d t-\int\left(\frac{d}{d t}(t) \cdot \int e^{t} d t\right) d t+\mathrm{C}$
$\Rightarrow y \cdot e^{\tan x}=t \cdot e^{t}-\int e^{\prime} d t+\mathrm{C}$
$\Rightarrow y e^{\tan x}=(t-1) e^{t}+\mathrm{C}$
$\Rightarrow y e^{\tan x}=(\tan x-1) e^{\tan x}+\mathrm{C}$
$\Rightarrow y=(\tan x-1)+\mathrm{C} e^{-\tan x}$
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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,