Solve the following PDE by separation of variables method , 3du/dx + 2du/dy = 0 , u(x,0) = 4e^-x
Answers
Answer:
Determine
for the following, simplifying your answer as far as possible.
(a) = (3+1)
sin−1()
(b) = tan−1 2
(c) = sin , = cos
(d) =
[Hint: Introduce logs]
2. Determine the equation of the tangent and normal of the following, giving your answer in exact form.
(a) − 2 + 2 = 4 at the point (0, 2)
(b) = 3( − sin ), = 3(1 − cos ) at =
4
3. Determine 2
2
for each of the following
(a) = cos(32 + 4)
(b) = + 2, = 22 + − 1
(c) = 3 , = 65
4. If = 32 − 2 + 5, show that 2 2
2 − 3
− 2 + 2(5 − 3) = 0
5. Determine the first and second order partial derivatives of
(a) = 34 + 53 + 6
(b) = 2+3
6. The function f is defined by the parametric equations
=
−
√1
−
2
and = sin−1 − 1 < ≤ 0.5
(a) Show that
= (
1−
2
)
2+
−
1
[6 marks]
(b) Hence show that f has no stationary value
[May 2015 P2]
7. A target is moving along a curve whose parametric equations are
= 4 − 3 cos , = 5 + 2 sin
where t is the time. The distances are measured in metres.
Let be the angle which the tangent to the cure makes with the positive x-axis.
(a) Determine the rate at which is increasing or decreasing when = 23
seconds.
[7 marks]
(b) What are the units of the rate of increase?
[1 mark]
(c) Determine the cartesian equation of the curve.
[2 marks]
[May 2011 P3/B]
8. (a)
Given that = cos−1 where 0 ≤ cos−1 ≤ , prove that
= − (
1
√1−
2 )
[7 marks]
(b)
The parametric equations of a curve are defined in terms of the parameter t by
= √1 − and = cos−1 ,wher
Answer:
Solve the following PDE by separation of variables method , 3du/dx + 2du/dy = 0 , u(x,0) = 4e^-x