\[
\textbf{Partial Differential Equations MCQs with Answers}
\]
\[
\textbf{Q1: Which of the following is the canonical form of a second-order linear PDE in two variables?}
\]
\[
\text{(A) } a \frac{\partial^2 u}{\partial x^2} + b \frac{\partial^2 u}{\partial x \partial y} + c \frac{\partial^2 u}{\partial y^2} + d \frac{\partial u}{\partial x} + e \frac{\partial u}{\partial y} + f u = 0
\]
\[
\text{(B) } u_{xx} + u_{yy} + u = 0
\]
\[
\text{(C) } u_{xx} – u_{yy} = 0
\]
\[
\text{(D) } \frac{\partial^2 u}{\partial x^2} + \frac{\partial u}{\partial x} = 0
\]
\[
\textbf{Answer: (A)}
\]
\[
\textbf{Q2: The equation } u_t + c u_x = 0 \textbf{ is an example of which type of PDE?}
\]
\[
\text{(A) } Parabolic \quad \text{(B) } Elliptic \quad \text{(C) } Hyperbolic \quad \text{(D) } None of these
\]
\[
\textbf{Answer: (C) Hyperbolic}
\]
\[
\textbf{Q3: For the Laplace equation } \nabla^2 u = 0, \textbf{ the solution satisfies which property?}
\]
\[
\text{(A) } Maximum principle \quad \text{(B) } Energy conservation \quad \text{(C) } Uniqueness only in bounded domains \quad \text{(D) } Symmetry of solutions
\]
\[
\textbf{Answer: (A) Maximum principle}
\]
\[
\textbf{Q4: The separation of variables method assumes the solution of a PDE can be written as:}
\]
\[
\text{(A) } u(x,y) = f(x)g(y) \quad \text{(B) } u(x,y) = f(x) + g(y) \quad \text{(C) } u(x,y) = f(x)g(x) \quad \text{(D) } u(x,y) = f(y) + g(x)
\]
\[
\textbf{Answer: (A) } u(x,y) = f(x)g(y)
\]
\[
\textbf{Q5: The wave equation } u_{tt} = c^2 u_{xx} \textbf{ is an example of a:}
\]
\[
\text{(A) } First-order PDE \quad \text{(B) } Second-order PDE \quad \text{(C) } Third-order PDE \quad \text{(D) } Fourth-order PDE
\]
\[
\textbf{Answer: (B) Second-order PDE}
\]
\[
\textbf{Q6: Which condition ensures the well-posedness of a PDE problem?}
\]
\[
\text{(A) } Existence, uniqueness, and continuous dependence on initial/boundary data
\]
\[
\text{(B) } Existence only \quad \text{(C) } Stability of solutions only \quad \text{(D) } None of these
\]
\[
\textbf{Answer: (A) Existence, uniqueness, and continuous dependence on initial/boundary data}
\]
\[
\textbf{Q7: In Fourier series solutions for PDEs, the orthogonality property is used to:}
\]
\[
\text{(A) } Find the coefficients of the series
\]
\[
\text{(B) } Determine the periodicity of the solution
\]
\[
\text{(C) } Ensure uniqueness of the solution
\]
\[
\text{(D) } Eliminate boundary conditions
\]
\[
\textbf{Answer: (A) Find the coefficients of the series}
\]
\[
\textbf{Q8: The method of characteristics is primarily used to solve:}
\]
\[
\text{(A) } Elliptic PDEs \quad \text{(B) } Hyperbolic PDEs \quad \text{(C) } Parabolic PDEs \quad \text{(D) } Integral equations
\]
\[
\textbf{Answer: (B) Hyperbolic PDEs}
\]
\[
\textbf{Q9: The Dirichlet boundary condition specifies:}
\]
\[
\text{(A) } Values of the function on the boundary
\]
\[
\text{(B) } Values of the derivative normal to the boundary
\]
\[
\text{(C) } The integral of the function over the domain
\]
\[
\text{(D) } None of these
\]
\[
\textbf{Answer: (A) Values of the function on the boundary}
\]
\[
\textbf{Q10: Which transformation is commonly used to solve PDEs in cylindrical coordinates?}
\]
\[
\text{(A) } Laplace transform \quad \text{(B) } Fourier transform \quad \text{(C) } Hankel transform \quad \text{(D) } Wavelet transform
\]
\[
\textbf{Answer: (C) Hankel transform}
\]