Discussion
Consider the differential equation \(x^{2}\frac{d^{2}y}{dx^{2}}+x\frac{dy}{dx}-4y=0\) with the boundary conditions of \(y(0)=0\) and \(y(1)=1\). The complete solution of the differential equation is
Answer: A
The solution is \(y = x^{2}\).
This is a Cauchy-Euler equation. Try \(y = x^{m}\):
\[ \begin{aligned} m(m-1)+m-4 &= 0 \\ m^{2} &= 4,\ m = \pm 2 \\ y &= c_{1}x^{2}+c_{2}x^{-2} \end{aligned} \]
The condition y(0) = 0 needs \(c_{2}=0\), since \(x^{-2}\) is unbounded at the origin, and y(1) = 1 then gives \(c_{1}=1\).
The three sine options satisfy both boundary values but not the equation: substituting \(\sin(\pi x/2)\), for example, leaves terms in \(x^{2}\sin\) and \(x\cos\) that do not cancel.
No comment is present. Be the first to comment.
Loading…
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