ENGG. MATHS-III-TRANSFORMS AND PDE-PRACTICE QUESTIONS FOR PDE

                                      PARTIAL DIFFERENTIAL EQUATIONS
                                       MODEL QUESTION PAPER
Marks:100                                                                                                                   Time:3hours
                                                    PART—A (16marks)
                               ATTEMPT ANY 10 OF THE FOLLOWING:(10*2=20):


01.Find the PDE of all planes having the equal intercepts on the x and y axis.


02.Find the complete solution of x2p2 + y2q2 = z2.


03.Find the general solution of
                                 4 ∂2z/∂x2 – 12∂2z/∂x∂y + 9∂2z/∂y2 = 0.


04.Eliminate the arbitrary constants and write down the PDE for
                                     (x –  a)2 + (y – b)2 = z2 cot2β


05.Solve the following PDE::Use direct integration method.
                                             ∂2z/∂x2 = a2z
    given that:   ∂z/∂x = asiny
                        ∂z/∂y = 0; when x=0;
06.Obtain the general, complete, singular solutions for the following PDE:
                        pq + p + q = 0.


07.Solve the following PDE by the method of grouping:
                               px + qy = z.


08.Solve the following higher order Partial differential equation:
∂2z/∂x2 – 6 ∂2z/∂y2 + ∂2z/∂x∂y = cos(3x + 4y).


09.Solve the following higher order non-homogenous PDE:
[(D – D' - 1)(D – D' – 2)]z = e2x-y.


10.Find the solution of the following partial differential equation:
y2p – yxq = x( z – 2y).


11.Form the PDE from the following function:
f ( y/x , x2+y2+z2 ) =0.


12.Find the complete integral of the following equation:
z/pq = x/q + y/p + √pq.


                                                   PART—B:(80 marks):
              ANSWER ANY 10 FROM THE FOLLOWING QUESTIONS:(10*8=80):


13.Solve the following equations:
(a) z = px + qy + √ p2 + q2 +1 (3+3+2 marks)
(b)p2 + q2 = x2 + y2
(c)z = p2 + q2.


14.Obtain the solutions of the following equations: (4+4 marks)
(a)x4p2 + y2zq = 2z2.
(b)x2p2 + xpq = z2.


15.Find the solutions of : (4+4marks)
(a)x4p2 – yzq = z2.
(b)z(p2 – q2) = x2 – y2.


16.Use the
(a) Langrange's multipliers
(b) method of grouping (5+3 marks)
to find the solution of :
(x2 + y2 +yz)p + (x2 +y2 – xz)q = z(x + y).


17.Solve the following:
(a) (D3 – 7DD'2 - 6D'3)z = sin(x+2y) + e2x+y . (5+3 marks)
(b) ∂2z/∂x2 + ∂2z/∂y2 = cos 2x cos 2y.


18.Obtain the solution of the following PDE: (5+3marks)
(a) (D2 – 6DD' + 5D'2)z = ex sinhy + xy.
(b)(D3 + D2D' – DD'2 – D'3)z = 3sin(x + y)


19.Find the solution of the following differential equations: (8 marks)
(2D4 – 3D2D' + D'2)z = e2x+y.


20.Solve each of the following: (5+3marks)
(a)D(D + D' – 1)(D +3D' – 2)z = e2x+3y .
(b)(D + 3D')(4D' + 3)z = cos(x + 4y).


21.Form the partial differential equations by eliminating the arbitrary functions
“ f “ and “ g “ and “Ψ” from the following: (2+2+2+2 marks)
(a)z = f( x3 + 2y) + g( x3 – 2y)
(b)z = x2 f(y) + y2 g(x)
(c)Ψ( x2 +y2 +z2 , xyz) = 0
(d)xy + z2 = Ψ(x + y +z).


22.(a)By changing the independent variables by the relations r = x+at , s = x-at,
show that the equation ∂2y/∂t2 = a2 ∂2y/∂x2 gets transformed to
∂2y/∂t ∂s = 0. hence find a general solution of the PDE.
(b) solve : z2 = p2 + q2 + 1. (5+3marks)




23.Solve the following PDE s : (4+2+2 marks)
(a)z2 (p2 + q2) = x2 + y2.
(b)z2 = xypq
(c)ap + bq + cz = 0.


24.Obtain the solutions of the following differential equations:
(a) (D2 – D'2 -3D')z = xy. (4+4marks)
(b) (D2 – DD' +D' -1)z = cos(x+2y) + ey-x.


25.Solve the following PDE completely using (3+5 marks)
(a)method of separation
(b)Multipliers
(x + 2z)p + (4xz – y)q = x2 +y .


ALL THE BEST AND BEST OF EFFORTS!!!


set by,
RANGARAJAN T.R.