EXAMPLES5
ORDINARY DIFFERENTIAL EQUATIONS
1998 1. Use the substitution where y and z are both functions of x to transform the differential equation
3y′ = exy−2 + y (1)
into
z′ = ex + z
and hence find the general solution of (1).
1994
1997
1999
2014, S2
2015, S1
2. A forced vibrating system is represented by
y′′ + 5y′ + 4y = 6sin(2t)
where 6sin(2t) is the driving force and y is the displacement from the equilibrium position. Find the motion of the system corresponding to the following initial displacement and velocity
y(0) = 1, y′(0) = 0.
Then find the steady state oscillations (i.e., the response of the system after a sufficiently long time).
3. Consider the differential equation
where c is a non-negative damping constant.
a) What damping constants c produce overdamping, critical damping, underdamping and no damping?
b) Sketch an example of the solution u(t) for the case of overdamping and for the case of underdamping.
4. Consider the vibrating system
.
Will the system in exhibit resonance for any choice of the forcing angular frequency ω? Give reasons for your answer.
5. Use the method of undetermined coefficients to solve the second order differential equation
y′′ + 2y′ + 5y = −25x2.
6. Use the method of undetermined coefficients to solve the second order differential equation
y′′ − 4y′ + 4y = 5sint.
2015, S2 7. Use the method of variation of parameters to find the general solution of the differential equation
.
.
y′′ + 3y′ + 2y = e−2t + 4t2 + 2.
Also describe the long term steady state solution.
.
y′′ − 4y = e2t.
.
y′′ + 9y = 6cos3t + 5et.
.
You are given that y1 = x and y2 = x3 are solutions to the corresponding homogeneous Euler–Cauchy ODE. You do not have to check this.
i) Calculate the Wronskian of y1 and y2.
ii) Use the method of Variation of Parameters to determine a particular solution yP for the inhomogeneous Euler–Cauchy ODE.
.
i) Find the solution yH to the homogeneous equation.
ii) For which value(s) of f will the system exhibit resonance? Give reasons for your answer.
(Note that you are not being asked to find the particular solution yP)
.
