Let y = y(x) be the solution of the differential equation x4 dy + (4x3 y + 2 x)dx = 0, x

Year	2026	2025	2024	Total
Questions	13	17	16	46

Year

2026

2025

2024

Total

Questions

Solution & Explanation

Related Formula

d(uv) = u dv + v du

Core Logic

The differential equation can be grouped using the exact differential logic.

x⁴ dy + (4x³ y + 2 x)dx = 0

(x⁴ dy + 4x³ y dx) = -2 x dx

Notice that $x⁴ dy + 4x³ y dx = d(x⁴ y)$ $x^4 dy + 4x^3 y dx = d(x^4 y)$. So, $\int d(x⁴ y) = \int -2 x dx$ $\int d(x^4 y) = \int -2\sin x dx$.

Step 1: Integrate both sides

Integrating gives:

x⁴ y = 2 x + C

Since $y(x) = f(x)$ $y(x) = f(x)$, we have $x⁴ f(x) = 2 x + C$ $x^4 f(x) = 2\cos x + C$.

Step 2: Find Integration Constant

Use the condition $f((π)/(2)) = 0$ $f\left(\frac{\pi}{2}\right) = 0$:

((π)/(2))⁴ (0) = 2 ((π)/(2)) + C \Rightarrow C = 0

So, the solution is: $x⁴ y = 2 x$ $x^4 y = 2\cos x$

Step 3: Final Calculation

Substitute $x = (π)/(3)$ $x = \frac{\pi}{3}$:

((π)/(3))⁴ y((π)/(3)) = 2 ((π)/(3))

(π⁴)/(81) y((π)/(3)) = 2((1)/(2)) = 1

π⁴ y((π)/(3)) = 81

Pattern Recognition

Whenever you see polynomial terms mixed with $xⁿ dy + n xⁿ⁻¹ y dx$ $x^n dy + n x^{n-1} y dx$, always condense it into the exact differential $d(xⁿ y)$ $d(x^n y)$ rather than using the standard integrating factor method.

Chapter Mix

Class 12 Maths: 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

Let $y = y(x)$ $y = y(x)$ be the solution of the differential equation $x⁴ dy + (4x³ y + 2 x)dx = 0$ $x^4 dy + (4x^3 y + 2\sin x)dx = 0$, $x > 0$ $x > 0$, $y((π)/(2)) = 0$ $y\left(\frac{\pi}{2}\right) = 0$. Then $π⁴ y((π)/(3))$ $\pi^4 y\left(\frac{\pi}{3}\right)$ is equal to:

Solution & Explanation

Related Formula

Core Logic

Step 1: Integrate both sides

Step 2: Find Integration Constant

Step 3: Final Calculation

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_2026_23_january_morning

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

Select an Exam

Let y = y(x)$y = y(x)$ be the solution of the differential equation x⁴ dy + (4x³ y + 2 x)dx = 0$x^4 dy + (4x^3 y + 2\sin x)dx = 0$, x > 0$x > 0$, y((π)/(2)) = 0$y\left(\frac{\pi}{2}\right) = 0$. Then π⁴ y((π)/(3))$\pi^4 y\left(\frac{\pi}{3}\right)$ is equal to:

Solution & Explanation

Related Formula

Core Logic

Step 1: Integrate both sides

Step 2: Find Integration Constant

Step 3: Final Calculation

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_2026_23_january_morning

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

Select an Exam

Welcome back

Let $y = y(x)$ $y = y(x)$ be the solution of the differential equation $x⁴ dy + (4x³ y + 2 x)dx = 0$ $x^4 dy + (4x^3 y + 2\sin x)dx = 0$, $x > 0$ $x > 0$, $y((π)/(2)) = 0$ $y\left(\frac{\pi}{2}\right) = 0$. Then $π⁴ y((π)/(3))$ $\pi^4 y\left(\frac{\pi}{3}\right)$ is equal to: