Discussion
Let \(D = \{(x, y) \in \mathbb{R}^{2} : x \gt 0\) and \(y \gt 0\}\). If the following second-order linear partial differential equation
\[ \begin{aligned} y^{2}\frac{\partial^{2}u}{\partial x^{2}} - x^{2}\frac{\partial^{2}u}{\partial y^{2}} \\ + y\frac{\partial u}{\partial y} = 0 \text{ on } D \end{aligned} \]
is transformed to
\[ \begin{aligned} &\left(\frac{\partial^{2}u}{\partial \eta^{2}} - \frac{\partial^{2}u}{\partial \xi^{2}}\right) + \left(\frac{\partial u}{\partial \eta} + \frac{\partial u}{\partial \xi}\right)\frac{1}{2\eta} \\ &+ \left(a\frac{\partial u}{\partial \eta} + b\frac{\partial u}{\partial \xi}\right)\frac{1}{2\xi} = 0 \text{ on } D \end{aligned} \]
for some \(a, b \in \mathbb{R}\), via the co-ordinate transform \(\eta = \frac{x^{2}}{2}\) and \(\xi = \frac{y^{2}}{2}\), then which one of the following is correct?
Answer: B
The coefficients are \(a = 0, b = -1\).
With \(\eta = x^{2}/2\), \(\xi = y^{2}/2\): \(u_{x} = xu_{\eta}\), \(u_{xx} = u_{\eta} + x^{2}u_{\eta\eta}\), \(u_{y} = yu_{\xi}\), \(u_{yy} = u_{\xi} + y^{2}u_{\xi\xi}\). Substituting:
\[ \begin{aligned} & x^{2}y^{2}(u_{\eta\eta} - u_{\xi\xi}) + y^{2}u_{\eta} \\ & - x^{2}u_{\xi} + y^{2}u_{\xi} = 0 \end{aligned} \]
Divide by \(x^{2}y^{2}\) and use \(x^{2} = 2\eta\), \(y^{2} = 2\xi\):
\[ \begin{aligned} &(u_{\eta\eta} - u_{\xi\xi}) + (u_{\eta} + u_{\xi})\frac{1}{2\eta} \\ &- u_{\xi}\frac{1}{2\xi} = 0. \end{aligned} \]
Comparing with the given form, \(a = 0\) and \(b = -1\). Any non-zero \(a\) would need a \(u_{\eta}/\xi\) term, which the computation does not produce.
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