Department of MathematicsIIT KHARAGPUR

Assignment-IMA41022 Partial Dierential Equaton

1. Eliminate the arbitrary function F from each of the following equationsand nd the corresponding PDE.

(a) F (xy; x+ y z) = 0(b) z = F

xyz

(c) z = F (x zy)

2. Eliminate the parameters a and b from the each of the following equationsand nd the corresponding PDE.

(a) z = (x+ a)(y + b)

(b) 2z = (ax+ y)2 + b

(c) z2(1 + a3) = 8(x+ ay + b)3

3. Find the general solution of

(a) y2p xyq = x(z 2y)(b) yzp+ xzq = x+ y

(c) 2x(y + z2)p+ y(2y + z2)q = z3

(d) (z2 2yz y2)p+ x(y + z)q = x(y z)4. Verify that the Pfaan dierential equations are integrable and nd the

corresponding integrals.

(a) (y2 + yz)dx+ (xz + z2)dy + (y2 xy)dz = 0(b) (1 + yz)dx+ x(z x)dy (1 + xy)dz = 0(c) (6x+ yz)dx+ (xz 2y)dy + (xy + 2z)dz = 0(d) (2x+ y2 + 2xz)dx+ 2xydy + x2dz = 0

5. Show that the equations p2+q21 = 0, (p2+q2)xpz = 0 are compatibleand nd the one parameter family of common solutions.

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