Let f:(0,infty)arrow R be a function which is differentiable at all points of its domain and satisfies the condition…

Year	2026	2025	2024	Total
Questions	13	17	16	46

Year

2026

2025

2024

Total

Questions

Solution & Explanation

Related Formula

The quotient rule derivative identity is given by:

(d)/(dx)((f(x))/(x²)) = (x² f'(x) - 2x f(x))/(x⁴)

Core Logic

Rearrange the given differential condition:

x² f'(x) - 2x f(x) = 3

Step 1: Divide by $x⁴$ $x^4$

To convert the left-hand side into an exact derivative form, divide the full relation by $x⁴$ $x^4$:

(x² f'(x) - 2x f(x))/(x⁴) = (3)/(x⁴)

(d)/(dx)((f(x))/(x²)) = 3x⁻⁴

Step 2: Integration and Evaluating Constant

Integrating both sides with respect to $x$ $x$:

(f(x))/(x²) = \int 3x⁻⁴ dx = -x⁻³ + C = -(1)/(x³) + C

f(x) = -(1)/(x) + Cx²

Using the given value $f(1) = 4$ $f(1) = 4$:

4 = -(1)/(1) + C(1)² \Rightarrow 4 = -1 + C \Rightarrow C = 5

Thus, the function is $f(x) = -(1)/(x) + 5x²$ $f(x) = -\frac{1}{x} + 5x^2$.

Step 3: Calculating $2f(2)$ $2f(2)$

Substitute $x = 2$ $x = 2$ to compute $2f(2)$ $2f(2)$ :

2 \times f(2) = 2 \times [ -(1)/(2) + 5(2)² ]

2 \times f(2) = 2 \times [ -(1)/(2) + 20 ] = -1 + 40 = 39

Pattern Recognition

Recognizing the structure $x² f'(x) - 2x f(x)$ $x^2 f'(x) - 2x f(x)$ as a partial quotient rule is faster than formatting it into standard linear order $(dy)/(dx) + P(x)y = Q(x)$ $\frac{dy}{dx} + P(x)y = Q(x)$ format, though both methods lead to the identical integration parameters safely.

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

Let $f:(0,\infty)arrow R$ $f:(0,\infty)\rightarrow R$ be a function which is differentiable at all points of its domain and satisfies the condition $x²f(x)=2xf(x)+3,$ $x^{2}f^{\prime}(x)=2xf(x)+3,$ with $f(1)=4$ $f(1)=4$ Then $2f(2)$ $2f(2)$ is equal to:

Solution & Explanation

Related Formula

Core Logic

Step 1: Divide by $x⁴$ $x^4$

Step 2: Integration and Evaluating Constant

Step 3: Calculating $2f(2)$ $2f(2)$

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

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Let f:(0,∞)arrow R$f:(0,\infty)\rightarrow R$ be a function which is differentiable at all points of its domain and satisfies the condition x²f(x)=2xf(x)+3,$x^{2}f^{\prime}(x)=2xf(x)+3,$ with f(1)=4$f(1)=4$ Then 2f(2)$2f(2)$ is equal to:

Solution & Explanation

Related Formula

Core Logic

Step 1: Divide by x⁴$x^4$

Step 2: Integration and Evaluating Constant

Step 3: Calculating 2f(2)$2f(2)$

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_24_jan_evening

Select Your Target Exam

Full Mock Test Hub

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Exam Analysis Tools

Chapter Weightage & Trends

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Formula Recognition

Exam Questions

Select an Exam

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Let $f:(0,\infty)arrow R$ $f:(0,\infty)\rightarrow R$ be a function which is differentiable at all points of its domain and satisfies the condition $x²f(x)=2xf(x)+3,$ $x^{2}f^{\prime}(x)=2xf(x)+3,$ with $f(1)=4$ $f(1)=4$ Then $2f(2)$ $2f(2)$ is equal to:

Step 1: Divide by $x⁴$ $x^4$

Step 3: Calculating $2f(2)$ $2f(2)$