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Math 423: Partial Differential Equation Ii Question Paper
Math 423: Partial Differential Equation Ii
Course:Bachelor Of Education Arts
Institution: Chuka University question papers
Exam Year:2013
CHUKA
UNIVERSITY
UNIVERSITY EXAMINATIONS
FOURTH YEAR EXAMINATIONS FOR THE AWARD OF
DEGREE OF BACHELOR OF EDUCATION ARTS
MATH 423: PARTIAL DIFFERENTIAL EQUATION II
STREAMS: BED (ARTS) Y4S2 TIME: 2 HOURS
DAY/DATE: MONDAY 22/4/2013 8.30 AM – 10.30AM
INSTRUCTIONS:
Answer Question ONE (Compulsory) and any other TWO Questions.
Adhere to the instructions on the question paper
Do not write on the Question Paper.
QUESTION ONE (30 MARKS)
(a) Given the second order partial differential equation as
A(x,y) U_xx+2B(x,y)Uxy+C(x,y) U_yy=0
Under what conditions it is classified as
(i) Hyperbolic
(ii) Parabolic
(iii) Elliptic [3 Marks]
(b) Classify the partial differential equation 6U_xx+14Uxy+4Uyy=0 and find its
characteristics. [4 Marks]
(c) Use the method of separation of variables to solve the differential equations below
(i) U_x=2U_t+U,given U(x,0)= ?6e?^(-3x) [5 Marks]
(ii) U_t=c^2 U_xx [6 Marks]
(d) (i) Define the term “Linear Partial differential equation [2 Marks]
(ii) State the order and linearity of the following differential equations, giving reasons
(a) (dy/dx)^2+y=x
(b) (d^2 y)/?dx?^2 +?4y?^2=0 [4 Marks]
(e) Determine whether the following p.d.e’s are hyperbolic, parabolic or elliptic using the
method of Characteristics. [6 Marks]
(i) The wave equation
U_tt=C^2 U_xx
(ii) Heat equation
U_t=?U_xx
(iii) Laplaced Equation
U_xx+U_yy=0
QUESTION 2 (20 MARKS)
(a) Give the D’Alembert’s solution of the one dimensional wave equation using the method
of characteristics.
U_tt=c^2 U_xx given that U(x,o)=?(x)and U_t (x,0)=?(x [13 Marks]
(b) Solve the partial differential equation
U_x=4U_y under the initial condition U(0,y)=?8e?^(-3y) [7 Marks]
QUESTION 3 (20 MARKS)
(a) Solve the wave equation
(d^2 U)/(dt^2 )=a^(2 ) (d^2 u)/?dx?^2 under the conditions U=0.When x=0 and x=p,
du/dt=0,when t=0 and U(x,0)=x where 0<x<p [15 Marks]
(b) Use the method of separation of variables to solve
2xdt/dx-3y dt/dy=0 [5 Marks]
QUESTION 4 (20 MARKS)
(a) State the general form of homogeneous linear partial differential equations of nth order with constant coefficient. [2 Marks]
(b) Given the equation
(d^2 z)/?dx?^2 -(?4d?^2 z)/dxdy+(?4d?^2 z)/?dy?^2 =0
(i) Find the auxiliary equation [1 Mark]
(ii) Solve the auxiliary equation and obtain the complementary Equation [2 Marks]
(c) Given the equation
(d^3 z)/?dx?^3 -(?3d?^3 z)/(?dx?^2 dy)+(?4d?^3 z)/?dy?^3 =e^(x+2y)
(i) Find the auxiliary equation [2 Marks]
(ii) Determine the complementary function. [1 Marks]
(iii) Find the particular integral [2 Marks]
(iv) Find the complete solution. [1 Mark]
(d) Find the general solution of equation
Z_xx+?3Z?_xy+?2z?_yy=x+y [9 Marks]
QUESTION 5 (20 MARKS)
(a) Given the following partial differential equation
(1+y) U_xx+2(1-x) U_xy+(1-y) U_yy=U((x,y,) determine the values of x and y
for which the equation is
(i) Hyperbolic
(ii) Parabolic
(iii) Elliptic [3 Marks]
(b) Given the P.d.e
3U_xx+?10U?_xy+?3U?_yy=0
(i) Find its characteristics [3 Marks]
(ii) Obtain the general solution by reducing it to the standard form [6 Marks]
(c) Using the method of variable separable, solve the two dimensional wave equation
U_tt=c^2 (U_xx+U_yy) [8 Marks]
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