Question

The second-order differential equation in an unknown function \( u: u(x,y) \) is defined as \[ \frac{\partial^2 u}{\partial x^2} = 2 \] Assuming \( g: g(x) \), \( f: f(y) \), and \( h: h(y) \), the general solution of the above differential equation is

• A. \( u = x^2 + f(y) + g(x) \)
• B. \( u = x^2 + x f(y) + h(y) \)
• C. \( u = x^2 + x f(y) + g(x) \)
• D. \( u = x^2 + f(y) + y g(x) \)

Hint:

When solving partial differential equations, always integrate with respect to the given variable and include arbitrary functions of other variables to account for the general solution.

Answer 1

The Correct Option is B

Step 1: The given differential equation is: \[ \frac{\partial^2 u}{\partial x^2} = 2 \] Integrating with respect to \( x \), we get: \[ \frac{\partial u}{\partial x} = 2x + C_1(y) \] where \( C_1(y) \) is an arbitrary function of \( y \). Step 2: Integrating again with respect to \( x \): \[ u = x^2 + x C_1(y) + C_2(y) \] where \( C_2(y) \) is another arbitrary function of \( y \). Step 3: From the given options, the correct representation of the general solution is: \[ u = x^2 + x f(y) + h(y) \] Conclusion: Thus, the correct answer is option (B) \( u = x^2 + x f(y) + h(y) \).