Download 2DE qns past papers

SECOND ORDER DIFFERENTIAL EQUATION EXAM QUESTIONS
FROM ‘OLD’ PAPERS
1.
(a) Show that y =
1
2
x2ex is a solution of the differential equation
d2 y
dy
2
+ y = ex.
2
dx
dx
(4)
(b) Solve the differential equation
d2 y
dy
2
+ y = ex.
2
dx
dx
given that at x = 0, y = 1 and
dy
= 2.
dx
(9)
[P4 January 2002 Qn 7]
2.
(a) Find the general solution of the differential equation
2
d2 y
dy
7
 3 y = 3t2 + 11t.
2
dt
dt
(8)
dy
(b) Find the particular solution of this differential equation for which y = 1 and
=1
dt
when t = 0.
(5)
(c) For this particular solution, calculate the value of y when t = 1.
(1)
[P4 June 2002 Qn 7]
3.
(a)
Find the value of  for which x cos 3x is a particular integral of the differential
equation
d2 y
+ 9y = 12 sin 3x.
dx 2
(4)
(b) Hence find the general solution of this differential equation.
(4)
dy
The particular solution of the differential equation for which y = 1 and
= 2 at x = 0, is
dx
y = g(x).
(c) Find g(x).
(d) Sketch the graph of y = g(x), 0  x  .
(4)
(2)
[P4 January 2003 Qn 7]
FP2 exam questions from past papers – Second Order Differential Equations
4.
d2 y
dy
 6 + 9y = 4e3t, t  0.
2
dt
dt
2 3t
(a) Show that Kt e is a particular integral of the differential equation, where K is a constant
to be found.
(4)
(b) Find the general solution of the differential equation.
(3)
dy
Given that a particular solution satisfies y = 3 and
= 1 when t = 0,
dt
(c) find this solution.
(4)
dy
Another particular solution which satisfies y = 1 and
= 0 when t = 0, has equation
dt
y = (1 – 3t + 2t2)e3t.
(d) For this particular solution draw a sketch graph of y against t, showing where the graph
crosses the t-axis. Determine also the coordinates of the minimum of the point on the
sketch graph.
(5)
[P4 June 2003 Qn 8]
5.
d2 y
dy
+4
+ 5y = 65 sin 2x, x > 0.
2
dx
dx
(a) Find the general solution of the differential equation.
(9)
(b) Show that for large values of x this general solution may be approximated by a sine
function and find this sine function.
(2)
[P4 January 2004 Qn 6]
6.
(a) Find the general solution of the differential equation
d2 y
dy
+2
+ 2y = 2et.
2
dt
dt
(6)
(b) Find the particular solution that satisfies y = 1 and
dy
= 1 at t = 0.
dt
(6)
[P4 June 2004 Qn 7]
FP2 exam questions from past papers – Second Order Differential Equations
7.
(a) Show that the transformation y = xv transforms the equation
x2
d2 y
dy
– 2x
+ (2 + 9x2)y = x5,
2
d
x
dx
I
into the equation
d 2v
+ 9v = x2.
2
dx
II
(5)
(b) Solve the differential equation II to find v as a function of x.
(6)
(c) Hence state the general solution of the differential equation I.
(1)
[FP1/P4 January 2005 Qn 6]
8.
(a) Find the general solution of the differential equation
2
d2x
dx
+5
+ 2x = 2t + 9.
2
dt
dt
(6)
(b) Find the particular solution of this differential equation for which x = 3 and
dx
= –1
dt
when t = 0.
(4)
The particular solution in part (b) is used to model the motion of a particle P on the x-axis. At
time t seconds (t  0), P is x metres from the origin O.
(c) Show that the minimum distance between O and P is
distance is a minimum.
1
2
(5 + ln 2) m and justify that the
(4)
[FP1/P4 June 2005 Qn 7]
9.
(a) Find the general solution of the differential equation
d2x
dx
+2
+ 5x = 0.
2
dt
dt
(4)
dx
= 1 at t = 0, find the particular solution of the differential
dt
equation, giving your answer in the form x = f(t).
(5)
(b) Given that x = 1 and
(c) Sketch the curve with equation x = f(t), 0  t  , showing the coordinates, as multiples
of , of the points where the curve cuts the t-axis.
(4)
[FP1/P4 January 2006 Qn 4]
FP2 exam questions from past papers – Second Order Differential Equations
10.
Given that 3x sin 2x is a particular integral of the differential equation
d2 y
+ 4y = k cos 2x,
dx 2
where k is a constant,
(a) calculate the value of k,
(4)
(b) find the particular solution of the differential equation for which at x = 0, y = 2, and for


which at x = , y = .
4
2
(4)
[FP1 June 2006 Qn 3]
11.
A scientist is modelling the amount of a chemical in the human bloodstream. The amount x of
the chemical, measured in mg l–1, at time t hours satisfies the differential equation
2
d2x
 dx 
–
6
  = x2 – 3x4, x > 0.
2
dt
 dt 
1
(a) Show that the substitution y = 2 transforms this differential equation into
x
2x
d2 y
+ y = 3.
dt 2
[I]
(5)
(b) Find the general solution of differential equation [ I ].
(4)
Given that at time t = 0, x 
1
dx
and
= 0,
2
dt
(c) find an expression for x in term of t,
(4)
(d) write down the maximum value of x as t varies.
(1)
[FP1 January 2007 Qn 7]
12.
For the differential equation
find the solution for which at x = 0,
d2 y
dy
+3
+ 2y = 2x(x + 3),
2
dx
dx
dy
= 1 and y = 1.
dx
(12)
[FP1 June 2007 Qn 5]
FP2 exam questions from past papers – Second Order Differential Equations
13.
(a)
Find the general solution of the differential equation
3
d 2 y dy
–
– 2y = x2.
2
d
x
dx
(8)
dy
(b) Find the particular solution for which, at x = 0, y = 2 and
= 3.
dx
(6)
[FP1 January 2008 Qn 7]
14.
(a)
Find, in terms of k, the general solution of the differential equation
d2x
dx
+4
+ 3x = kt + 5,
2
dt
dt
where k is a constant and t > 0.
(7)
For large values of t, this general solution may be approximated by a linear function.
(b) Given that k = 6, find the equation of this linear function.
(2)
[FP1 June 2008 Qn 4]
15.
(x2 + 1)
d2 y
dy
= 2y2 + (1 – 2x) .
2
dx
dx
(I)
(a) By differentiating equation (I) with respect to x, show that
(x2 + 1)
d3 y
d2 y
dy
=
(1
–
4x)
+ (4y – 2) .
3
2
dx
dx
dx
(3)
dy
Given that y = 1 and
= 1 at x = 0,
dx
(b) find the series solution for y, in ascending powers of x, up to and including the term in x3.
(4)
(c) Use your series to estimate the value of y at x = –0.5, giving your answer to two decimal
places.
(1)
[FP3 June 2008 Qn 3]
FP2 exam questions from past papers – Second Order Differential Equations