Discussion
The function \(\phi(x)\) satisfying the integral equation
\[ \int_{0}^{x} e^{x - \xi} \phi(\xi)\, d\xi = \frac{x^{2}}{2} \]
is
Answer: C
The solution is \(\phi(x) = x - \frac{x^{2}}{2}\).
The left side is the convolution of \(e^{x}\) with \(\phi\), so take Laplace transforms:
\[ \begin{aligned} \frac{1}{s - 1}\,\Phi(s) &= \frac{1}{s^{3}} \\ \Phi(s) &= \frac{s - 1}{s^{3}} = \frac{1}{s^{2}} - \frac{1}{s^{3}} \\ \phi(x) &= x - \frac{x^{2}}{2} \end{aligned} \]
Check by differentiating the equation: \(\phi(x) + \int_{0}^{x} e^{x-\xi}\phi(\xi)\,d\xi = x\), so \(\phi(x) = x - x^{2}/2\) directly. \(x^{2}/2\) mistakes the right side for the answer, and \(x + x^{2}/2\) and \(1 + x^{2}/2\) have the wrong sign or fail at \(x = 0\), where \(\phi\) must satisfy \(\phi(0) = 0\).
Post your comment
Please log in or register to post a comment.
Examination Categories
- SSC Exams
- Banking Exams
- Teaching Exams
- Civil Services Exams
- Railways Exams
- Engineering Recruitment Exams
- Defence Exams
- State Government Exams
- Police Exams
- Insurance Exams
- Nursing Exams
- Other Government Exams
- NRA CET
- PG Entrance Exams
- Campus Placement
- MBA Entrance Exams
- CUET and UG Entrance
- Judiciary Exams
- Regulatory Body Exams
- Software Development