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Question #41471

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x, t)$ be a continuously differentiable function for $x > 0$ and $t \ge 0$ satisfying the integral equation: $\int_0^1 \left( f(x, yt) + x \frac{\partial f}{\partial x}(x, yt) \right) dy = x^2 \frac{e^t - 1}{t} + x (t + 2)$ for all $x > 0$ and $t > 0$ . If $f(x, 0) = \frac{x^2}{3} + x$ and $\frac{\partial f}{\partial t}(1, t) = \frac{e^t}{3} + 2t + 1$ , find the value of $f(3, 3)$ .

Options

A.
$3 e^3 + 15$
B.
$3 e^3 + 12$
C.
$3 e^3 + 14$
D.
$9 e^3 + 45$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Integrating f(x) + xf'(x) forms|Leibniz Rule for Differentiation under Integral|Integration Constant Trap

Why This Question Is Tricky

Common Mistakes

This question tests the application of the Leibniz rule and the integration of exact differentials, specifically targeting the 'Integration Constant Trap'.

When integrating a multivariable function with respect to one variable (here, $x$ ), the constant of integration is a function of the other variable ( $C(t)$ ).

Students frequently treat this constant as a fixed numerical value.

If a student assumes $C$ is a constant, they will use the initial condition $f(x, 0)$ to find $C=0$ , completely ignoring the second boundary condition, leading to the incorrect answer $3e^3 + 12$ .

The problem forces the realization that $C(t)$ is a function by providing a differential condition $\frac{\partial f}{\partial t}(1, t)$ that yields $C'(t) = 2t$ , requiring a second integration to find $C(t) = t^2$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Integrals - Class 12
Concept	Integrating f(x) + xf'(x) forms
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free