Discussion
The solution of the integral equation
\[ x(t) = t + \int_{0}^{t} (s - t)\, x(s)\, ds \]
with the help of resolvent kernel is :
Answer: B
The solution is \(x(t) = \sin t\).
The kernel is \(s - t = -(t - s)\). The resolvent route gives the same result as differentiating, which is quicker:
\[ \begin{aligned} x(t) &= t - \int_{0}^{t} (t - s)\, x(s)\, ds \\ x'(t) &= 1 - \int_{0}^{t} x(s)\, ds \\ x''(t) &= -x(t) \end{aligned} \]
with \(x(0) = 0\) and \(x'(0) = 1\). The solution is \(\sin t\). (The resolvent kernel of \(-(t - s)\) is \(-\sin(t - s)\), and \(x = t - \int_{0}^{t}\sin(t - s)\, s\, ds = \sin t\).)
\(\cos t\), \(e^{t}\) and 1 all fail \(x(0) = 0\), which the equation forces at \(t = 0\) because the integral vanishes there.
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