Partial Differential Equations Questions and Answers

Answer: A

The discriminant \( B^{2} - 4AC \) is \( -4 \), zero and \( 4c^{2} \) respectively. Negative gives elliptic, with data round a closed boundary; zero gives parabolic, marching forward from an initial state; positive gives hyperbolic, carrying disturbances along characteristics at finite speed

Answer: D

Lagrange's linear form is integrated through its subsidiary equations. Clairaut's form needs nothing more than replacing the two derivatives by constants. Anything outside the standard types goes to Charpit's method, which is general but longer

Answer: C

The three coefficients are read off first, then written as the subsidiary equations. Two independent integrals \( u = a \) and \( v = b \) follow, and the general solution is \( \phi(u,v) = 0 \) for an arbitrary \( \phi \). Writing the solution before the integrals are found leaves nothing to put inside the function

Answer: B

All three hold. The presence of arbitrary functions is the essential difference from the ordinary case, and it is why one boundary curve rather than a few numbers is needed to pin a solution down. The discriminant test mirrors the classification of conics, and for the same algebraic reason

Answer: A

The first two hold. The third fails: two initial conditions, the initial displacement and the initial velocity, need two arbitrary functions to satisfy, so one travelling wave is not enough unless the two conditions happen to be compatible with it. That is exactly why d'Alembert's solution carries both terms

Answer: D

Linearity requires the unknown and its derivatives to appear only to the first power and never multiplied by one another; coefficients depending on \( x \) and \( y \) are unrestricted. The squares and the products \( pq \) each break that, and it is those cases that need the nonlinear machinery of the standard types and Charpit's method

Answer: C

The complete integral of a first-order equation in two independent variables carries two arbitrary constants, one for each variable, and the general solution replaces them by an arbitrary function. Counting one constant per order is the ordinary-equation habit, and it does not carry across

Answer: B

If \( A \), \( B \) and \( C \) depend on \( x \) and \( y \), the discriminant does too, and it may change sign. The Tricomi equation \( y z_{xx} + z_{yy} = 0 \) is elliptic for \( y > 0 \) and hyperbolic for \( y < 0 \), a mixed type that arises physically at transonic speeds. Linearity is not what decides it