The partial differential equation \[ 7 \frac{\partial^2 u}{\partial x^2} + 16 \frac{\partial^2 u}{\partial x \partial y} + 4 \frac{\partial^2 u}{\partial y^2} = 0 \] is transformed to \[ A \frac{\partial^2 u}{\partial \xi^2} + B \frac{\partial^2 u}{\partial \xi \partial \eta} + C \frac{\partial^2 u}{\partial \eta^2} = 0, \] using \( \xi = y - 2x \) and \( \eta = 7y - 2x \). Then, the value of \( \frac{1}{123} (B^2 - 4AC) \) is _____
The partial differential equation \[ 7 \frac{\partial^2 u}{\partial x^2} + 16 \frac{\partial^2 u}{\partial x \partial y} + 4 \frac{\partial^2 u}{\partial y^2} = 0 \] is transformed to \[ A \frac{\partial^2 u}{\partial \xi^2} + B \frac{\partial^2 u}{\partial \xi \partial \eta} + C \frac{\partial^2 u}{\partial \eta^2} = 0, \] using \( \xi = y - 2x \) and \( \eta = 7y - 2x \). Then, the value of \( \frac{1}{123} (B^2 - 4AC) \) is _____
Correct Answer: 12
Solution and Explanation
Partial Differential Equation Transformation
The given partial differential equation is:
\( 7 \frac{\partial^2 u}{\partial x^2} + 16 \frac{\partial^2 u}{\partial x \partial y} + 4 \frac{\partial^2 u}{\partial y^2} = 0 \)
The transformation to new coordinates \( \xi = y - 2x \) and \( \eta = 7y - 2x \) is required. We need to find the value of \( \frac{1}{123} (B^2 - 4AC) \) after transforming the equation.
Step 1: Compute the Jacobian Matrix
The partial derivatives of \( \xi \) and \( \eta \) with respect to \( x \) and \( y \) are:
\( \frac{\partial \xi}{\partial x} = -2, \quad \frac{\partial \xi}{\partial y} = 1 \)
\( \frac{\partial \eta}{\partial x} = -2, \quad \frac{\partial \eta}{\partial y} = 7 \)
The inverse Jacobian matrix is:
\( \frac{\partial x}{\partial \xi} = -\frac{1}{2}, \quad \frac{\partial y}{\partial \xi} = \frac{1}{2}, \quad \frac{\partial x}{\partial \eta} = -\frac{1}{2}, \quad \frac{\partial y}{\partial \eta} = \frac{7}{2} \)
Step 2: Transformation of the Partial Derivatives
We transform the second-order partial derivatives of \( u \) with respect to \( x \) and \( y \) into those with respect to \( \xi \) and \( \eta \) using the chain rule. We express the equation as:
\( A \frac{\partial^2 u}{\partial \xi^2} + B \frac{\partial^2 u}{\partial \xi \partial \eta} + C \frac{\partial^2 u}{\partial \eta^2} = 0 \)
Using the matrix form for the coefficients of the PDE:
\[ \begin{pmatrix} A & B \\ B & C \end{pmatrix} = \begin{pmatrix} 7 & 16 \\ 16 & 4 \end{pmatrix} \]
Step 3: Compute \( B^2 - 4AC \)
The determinant \( B^2 - 4AC \) is computed as follows:
\( B = 16, \quad A = 7, \quad C = 4 \)
Now calculate \( B^2 - 4AC \):
\[ B^2 - 4AC = 16^2 - 4(7)(4) = 256 - 112 = 144 \]
Step 4: Final Answer
Thus, the value of \( \frac{1}{123} (B^2 - 4AC) \) is:
\[ \frac{1}{123} (B^2 - 4AC) = \frac{1}{123} \times 144 = \frac{144}{123} = \frac{12}{1} \]
So, the value is \( \boxed{12} \).
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