If for the solution curve y = f(x) of the differential equation (dy/dx) + ( x)y = (2 + x/(1 + 2 x)2), x ((-pi/2)…

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Year

2026

2025

2024

Total

Questions

Solution & Explanation

Related Formula

Integrating factor (I.F.) for a linear differential equation $(dy)/(dx) + Py = Q$ $\frac{dy}{dx} + Py = Q$:

I.F. = e\int P dx

General solution:

y \cdot (I.F.) = \int Q \cdot (I.F.) dx

Core Logic

Given $P = x$ $P = \tan x$, compute Integrating Factor:

I.F. = e\int x dx = eln( x) = x

Set up integrated expression solution layout:

y \cdot x = \int (2 + x)/((1 + 2 x)²) \cdot x dx = \int (2 x + 1)/(( x + 2)²) \cdot dx

Step 1: Evaluate Integration with Half-Angle Substitutions

Using tangent half-angle substitution $t = (x)/(2)$ $t = \tan\frac{x}{2}$ transformations simplifies the integral loop structure down to:

y \cdot x = (2)/(t + (3)/(t)) + C

Plugging entry condition parameters $f((π)/(3)) = \sqrt(3)10$ tracking $t = 1\sqrt(3)$ $t = \frac{1}{\sqrt{3}}$ explicitly isolates boundary condition constant $C$ $C$: $C = 0$ $C = 0$

Step 2: Calculate Target Point Value

At target query point $x = (π)/(4)$ $x = \frac{\pi}{4}$, half-angle parameters scale to $t = \sqrt(2) - 1$ $t = \sqrt{2} - 1$:

y \cdot \sqrt(2) = 2\sqrt(2) - 1 + 3\sqrt(2) - 1 = 2(\sqrt(2) - 1)6 - 2\sqrt(2)

y = 4 - \sqrt(2)14

Pattern Recognition

When integrating complex rational expressions involving trigonometric values, half-angle substitution methods ( $t = (x)/(2)$ $t = \tan\frac{x}{2}$) are standard for reducing polynomial degrees.

Chapter Mix

Class 12 Mathematics: Differential Equations

More Differential Equations Previous-Year Questions — Page 10

Q11 jee_main_2024_31_jan_morning Linear Differential Equations

Let

y = y(x)

be the solution of the differential equation

(dy)/(dx) = (( x) + y)/( x( x - x x)), x in (0, (π)/(2))

satisfying the condition

y((π)/(4)) = 2

. Then,

y((π)/(3))

A. $\sqrt(3)(2 + ₑ\sqrt(3))$ $$\sqrt{3}(2 + \log_e\sqrt{3})$$
B. $\sqrt(3)2(2 + ₑ 3)$ $$\frac{\sqrt{3}}{2}(2 + \log_e 3)$$
C. $\sqrt(3)(1 + 2 ₑ 3)$ $$\sqrt{3}(1 + 2\log_e 3)$$
D. $\sqrt(3)(2 + ₑ 3)$ $$\sqrt{3}(2 + \log_e 3)$$

Solution

Core Logic

(dy)/(dx) = (( x)/( x) + y)/( x ((1)/( x) - ( ² x)/( x))) = ( x + y x)/( x (1 - ² x))

(dy)/(dx) = ( x + y x)/( x ² x) = ² x + (2y)/( 2x)

(dy)/(dx) - 2 (2x)y = ² x

Step 1: Integrating Factor

This is an LDE of form $(dy)/(dx) + Py = Q$ $\frac{dy}{dx} + Py = Q$.

I.F. = e\int -2 (2x) dx

Let $2x = t 2dx = dt$ $2x = t \implies 2dx = dt$.

I.F. = e-\int t dt = e-ln| (t/2)| = e-ln| x| = (1)/(| x|)

Step 2: Solution of LDE

y(I.F.) = \int Q(I.F.) dx + C

y(1)/( x) = \int ² x (1)/( x) dx + C

Let $x = t ² x dx = dt$ $\tan x = t \implies \sec^2 x dx = dt$.

y(1)/( x) = \int (dt)/(t) + C = ln| x| + C

y = x(ln| x| + C)

Step 3: Boundary Value

Given $y(π/4) = 2$ $y(\pi/4) = 2$:

2 = 1(ln 1 + C) C = 2

Thus, $y = x (ln| x| + 2)$ $y = \tan x (\ln|\tan x| + 2)$. At $x = π/3$ $x = \pi/3$:

y(π/3) = \sqrt(3)(ln\sqrt(3) + 2)

Chapter Mix

Class 12 Maths: Differential Equations

If for the solution curve $y = f(x)$ $y = f(x)$ of the differential equation $(dy)/(dx) + ( x)y = (2 + x)/((1 + 2 x)²)$ , $x in ((-π)/(2), (π)/(2)), f((π)/(3)) = \sqrt(3)10$ , then $f((π)/(4))$ $f\left(\frac{\pi}{4}\right)$ is equal to:

Solution & Explanation

Related Formula

Core Logic

Step 1: Evaluate Integration with Half-Angle Substitutions

Step 2: Calculate Target Point Value

Pattern Recognition

Chapter Mix

Reference Study Guides

More Differential Equations Previous-Year Questions — Page 10

Solution

Core Logic

Step 1: Integrating Factor

Step 2: Solution of LDE

Step 3: Boundary Value

Chapter Mix

More Differential Equations Questions — jee_main_2025_29_jan_evening

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Solution & Explanation

Related Formula

Core Logic

Step 1: Evaluate Integration with Half-Angle Substitutions

Step 2: Calculate Target Point Value

Pattern Recognition

Chapter Mix

Reference Study Guides

More Differential Equations Previous-Year Questions — Page 10

Solution

Core Logic

Step 1: Integrating Factor

Step 2: Solution of LDE

Step 3: Boundary Value

Chapter Mix

More Differential Equations Questions — jee_main_2025_29_jan_evening

Select Your Target Exam

Full Mock Test Hub

Topic-wise Practice Hub

Exam Analysis Tools

Chapter Weightage & Trends

Topic-wise Practice Hub

JEE Main Exam Papers

Chapter Weightage Board

Formula Recognition

Exam Questions

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