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Math 420: Partial Differential Equations I Question Paper
Math 420: Partial Differential Equations I
Course:Bachelor Of Science
Institution: Chuka University question papers
Exam Year:2013
CHUKA
UNIVERSITY
UNIVERSITY EXAMINATIONS
FOURTH YEAR EXAMINATION FOR THE AWARD OF DEGREE OF
BACHELOR OF SCIENCE
MATH 420: PARTIAL DIFFERENTIAL EQUATIONS I
STREAMS: BSC (MATHEMATICS) Y4S1 TIME: 2 HOURS
DAY/DATE: THURSDAY 15/8/2013 11.30 A.M. – 1.30 P.M.
INSTRUCTIONS:
Answer Question One (Compulsory) and two questions.
Adhere to the instructions on the answer booklet.
Do not write on the question paper.
QUESTION ONE (COMPULSORY)
Define the following terms:
An ordinary differential equation [1 mark]
A partial differential equation [1 mark]
State the linearity, order and degree of the following partial differential equations
giving reasons.
1+?(Uxx)?^2=Uxy [2 marks]
?Uxxy+2e?^x Uyy+UxyUy=x^4 [2 marks]
Given that U=?Sin?^(-1) (x-y),x=3t,y=?4t?^3, Calculate du/dt . [3 marks]
Use Jacobians to find ?U/?x if U^2+V^2-xy=0,U^2+xyV+V^2=0. [4 marks]
Find the equations of the tangent plane and normal line to the surface
?2x?^2+y^2+22=3 at the point (2, 1, -3). [4 marks]
Solve the Langrange’s linear equation
?(x?^2-yz)p+?(y?^2-zx)q=z^2-xy {4 marks]
Find the integral surface of the linear partial differential equation
xp+yq=zwhich contains the circle defined by x^2+y^2+z^2=4
andx+y+z=2. [6 marks]
Determine the orthogonal trajectories of the family of sinusoids.
y=C Sinx [3 marks]
QUESTION TWO
Solve the simultaneous differential equations
dx/dt-dy/dt+2y=Cos 2t
dx/dt+dy/dt-2x=Sin 2t [10 marks]
Solve the Jacobi differential equation
(P_1+P_2 ) P_(3 ) z+x+y=0 [8 marks]
Define Pffafian differential equation and state the necessary condition for it to be integrable. [2 marks]
QUESTION THREE
Solve the differential equation
(?^2 z)/?x?y=x^2 ysubject to
z(x,0)=x^2andz(1,y)=Cos y by direct integration. [5 marks]
Verify that the Pffafian equation
(y+z)dx+(x+z)dy+(x+y)dz=0isintegrable, Hence solve it. [5 marks]
A family of hyperbolic curves is given the equation
y=C/x
Find the orthogonal trajectories for these curves. [5 marks]
Find an integral surface of xp+yq=z passing through the curve
x^2+y^2=1,x^2+y^2+z^2=25 [5 marks]
QUESTION FOUR
Solve the partial differential equation
?(p?^2+q^2) y=qzusing Charpits method. [15 marks]
Using the method of characteristics, solve the equation
?U/??x?_1 (x_1 x_2 )+x_1 ?U/??x?_2 (x_1 x_2 )=0. [5 marks]
QUESTION FIVE
Find the equation of the tangent plane and normal line to the surface
xyz=6at (1 2 3). [3 marks]
Solve the Langrange’s linear differential equation
x^2 dz/dx+y^2 dz/dy=(x+y)z [4 marks]
Given that U=Sin (x^2+y^2) and y=f(x)
where a^2 x^2+b^2 y^2=c^2 , find du/dx. [6 marks]
Solve the simultaneous differential equations below
dx/dt+2x-3=0
dy/dt-3x+2y=0 [7 marks]
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