Let f be a twice differentiable non-negative function such that (f(x))2 = 25 + 0x ( (f(t))2 + (f'(t))2 ) dt. Then the…

Year	2026	2025	2024	Total
Questions	13	17	16	46

Year

2026

2025

2024

Total

Questions

Solution & Explanation

Related Formula

(d)/(dx)\int₀^x g(t)dt = g(x) (Leibniz Rule)

Core Logic

Differentiate both sides with respect to $x$ $x$ using Leibniz rule:

2f(x)f'(x) = f²(x) + (f'(x))²

Rearranging yields a perfect square:

(f'(x))² - 2f(x)f'(x) + f²(x) = 0

(f'(x) - f(x))² = 0

Thus, $f'(x) = f(x)$ $f'(x) = f(x)$.

Step 1: Solve the Differential Equation

Integrate $(f'(x))/(f(x)) = 1$ $\frac{f'(x)}{f(x)} = 1$:

ln f(x) = x + C

f(x) = ex+C = k e^x

To find $k$ $k$, plug $x = 0$ $x = 0$ into the original integral equation:

(f(0))² = 25 + \int₀⁰ dt = 25

Since $f$ $f$ is non-negative, $f(0) = 5$ $f(0) = 5$. Thus, $k = 5$ $k = 5$ and $f(x) = 5e^x$ $f(x) = 5e^x$.

Step 2: Calculate Sequence Values

Evaluate $f(ln n)$ $f(\ln n)$ for $n=1, 2, \dots, 625$ $n=1, 2, \ldots, 625$:

f(ln n) = 5eln n = 5n

Step 3: Calculate the Mean

Mean of $f(ln 1), f(ln 2), , f(ln 625)$ $f(\ln 1), f(\ln 2), \dots, f(\ln 625)$:

Mean = Σn=1⁶²⁵ 5n625 = (5 (1 + 2 + + 625))/(625)

= (5)/(625) \times (625 \times 626)/(2) = (5 \times 626)/(2) = 5 \times 313 = 1565

Pattern Recognition

An integral equation with $x$ $x$ as an upper limit immediately demands Leibniz differentiation. If it results in a perfect square differential relation, an exponential solution map natively emerges.

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 $f$ $f$ be a twice differentiable non-negative function such that $(f(x))² = 25 + \int₀x ( (f(t))² + (f'(t))² ) dt.$ Then the mean of $f( ₑ(1))$ $f(\log_{e}(1))$, $f( ₑ(2))$ $f(\log_{e}(2))$, ..., $f( ₑ(625))$ $f(\log_{e}(625))$ is equal to _____.

Solution & Explanation

Related Formula

Core Logic

Step 1: Solve the Differential Equation

Step 2: Calculate Sequence Values

Step 3: Calculate the Mean

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

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Solution & Explanation

Related Formula

Core Logic

Step 1: Solve the Differential Equation

Step 2: Calculate Sequence Values

Step 3: Calculate the Mean

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

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