Question

Consider a system of the following two partial differential equations :
\(\frac{∂\alpha}{∂x}=-2\frac{∂\beta}{∂t}\)
\(\frac{∂\beta}{∂x}=-2\frac{∂\alpha}{∂t}\)
Which one of the following choices is a possible solution for the system ?

• A. α(t, x) = (x - t)² + (x + t)² and β(t, x) = (x - t)² - (x + t)².
• B. α(t, x) = (x - 2t)² + (x +2t)² and β(t, x) = (x - 2t)² - (x + 2t)².
• C. \(\alpha(t,x)=(x-\frac{t}{2})^2+(x+\frac{t}{2})^2\ \text{and}\ \beta(t,x)=(x-\frac{t}{2})^2-(x+\frac{t}{2})^2\)
• D. \(\alpha(t,x)=(x-\frac{t}{2})^2+2(x+\frac{t}{2})^2\ \text{and}\ \beta(t,x)=2(x-\frac{t}{2})^2-(x+\frac{t}{2})^2\)

Answer 1

The Correct Option is C

The correct option is (C) : \(\alpha(t,x)=(x-\frac{t}{2})^2+(x+\frac{t}{2})^2\ \text{and}\ \beta(t,x)=(x-\frac{t}{2})^2-(x+\frac{t}{2})^2\).