Mathematics for Physics 3: Dynamics and Differential Equations
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Mathematics for Physics 3: Dynamics – Worksheet 6
Preparation problems
You are expected to do the following questions before your tutorial workshop, as a preparation.
Consulting friends as well as the course team is encouraged.
1. Find the general solution of the following equations for x(t):
A. xẍ − ẋ2 = 0.
B. xẍ + ẋ2 = 0.
C. ẍ + k ẋ2 = 0. In this case find the solution for x(0) = 0.
2. D’Alembert method (reduction of other):
A. Find the general solution of
B. Verify that y = xe
−x
dv
dx
− 2v(x) = 1.
satisfies the following homogeneous equation for y(x):
x2 y �� − 2xy � + (2 − x2 )y = 0
C. By D’Alembert method find the general solution of
x2 y �� − 2xy � + (2 − x2 )y = x3 e−x
using the substitution: y(x) = x e−x u(x).
3. Using D as a differential operator,
A. Prove the following identity
B. show that
and hence that
�
�
(D − α)(D − β) f (x)eλx = eλx (D + λ − α)(D + λ − β)f (x)
�
�
� �
(D + 1)(D − 3) xn e3x = e3x (D + 4)D xn
�
�
(D + 1)(D − 3) xn e3x = e3x xn−2 n(n − 1 + 4x)
C. Use B above to find a particular solution yP (x) for the following linear differential equation:
y �� − 2y � − 3y = −16xe3x
Hint: Take a trial function (Ax2 + Bx)e3x and use n = 1 and n = 2 in the final result in B.
4. Solve the following differential equation for y(x):
y �� − 4y � + 3y = 1 + 3x
By separating into homogeneous and non-homogeneous contributions and substituting a finite polynomial for
the non-homogeneous part and y(x) = eαx for the homogeneous part.
Workshop problems
You are expected to do the following questions during the tutorial workshop.
Consulting friends as well as the course team is encouraged.
1. In each of the following cases verify that u(x) is a solution of the homogeneous equation Ô[y] = 0 for y(x), and
hence find the general solution for the corresponding inhomogeneous equation Ô[y] = f (x). Use D’Alembert
method to reduce the equation to first order, and then use an integrating factor to solve the latter.
A. u(x) = sin(x), Ô = D2 + 1, f (x) = sin(2x) .
B. u(x) = x2 , Ô = x2 D2 − 2, f (x) = x3 .
2
d
d
Notation: D denotes differentiation with respect to x, i.e. D = dx
, and D2 = dx
2 ; a number in Ô should be
interpreted as that number time the identity operator, for example (D2 + 1)[y] = y �� + y.
N.B. You are requested to apply the method described above even if you know a faster way to obtain the final
result. Alternative derivations should only be used to check your result.
Mathematics for Physics 3: Dynamics and Differential Equations
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2. For each of the following second order linear ODE’s, write the most general solution for y(x). To determine the
particular non-homogeneous solution, you have to chose between applying the D’Alembert method or a trial
function.
3
A. y �� − 4y = sin(3x) . If y(0) = − 26
and y � (0) = 0, what is the steady state solution?
B. y �� + y � − 2y = 14 + 2x − 2x2 . In this case what is the solution given y(0) = 0 and y � (0) = 0?
C. y �� − y � − 2y = 10 sin(x) .
D. y �� − y � − 2y = 3e2x .
3. Determine the general solution y(x) of the following equation
y� − y = 1
using a series substitution
y(x) = a0 + a1 x + a2 x2 + · · · =
Check your solution by solving the equation directly.
∞
�
an xn
n=0
4. Given the following differential equation for y(x):
y (4) − 2y (3) + 2y (2) − 2y � + y = 5 + 43 x2 ex
where y (n) represents the n-th derivative of y(x),
A. Substitute y = eαx and derive the characteristic equation. Next, find the roots of the characteristic
equation and determine the general solution of the homogeneous equation.
B. Use the D’Alembert method to find a particular non-homogeneous solution.
5. Let Ô be a linear differential operator of order n,
Ô =
n
�
gi (t) Di
i=1
d
where D = dt
.
Given that the functions uk (t) (for k = 1 through n), ya (t) and yb (t) satisfy
Ô [uk (t)] = 0 , ∀k ;
�
�
n
�
Ô ya (t) +
Ak uk (t) = 3f1 (t) + 2f2 (t) ;
�
k=1
Ô yb (t) +
n
�
k=1
�
Bk uk (t) = f2 (t) − f1 (t) ,
where Ak and Bk are constants, find the general solution y(t) of the equation:
Ô[y(t)] = αf1 (t) + βf2 (t)
Mathematics for Physics 3: Dynamics and Differential Equations
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Hand-in problems
The following questions must be solved independently by each student and handed in for assessment: consultation
is not allowed!
1. For each of the following second order linear ODE’s, write the most general solution for y(x). To determine the
particular non-homogeneous solution, you have to chose between applying the D’Alembert method or a trial
function.
A. y �� − 2y � + 2y = 2x + 3 .
B. y �� − 2y � + y = ex ln(x) .
C. 2y �� − 2y � + y = 2e−x .
D. 3y �� − 2y � − y = e2x + 3 . In this case what is the solution given y(0) = 0 and y � (0) = 0?
2. Solution by series:
A. Solve the following differential equation
y � + (tan(x))y = cos(x)
by a power series substitution,
y(x) = a0 + a1 x + a2 x2 + · · · =
∞
�
an xn
n=0
Check your solution by solving it again using an integrating factor and by substitution.
Guidance: Determine the first the three (non-zero) terms in the Maclaurin expansion (Taylor series
around x = 0) of tan(x) and cos(x). Then expand each term in the equation above, and compare the
coefficients of the various power of x on both sides. Once you known the coefficients a1 through a4 (in
terms of a0 ) guess the functional form. Finally, check it as instructed above.
B. Using a series substitution:
u(x) = a0 + a1 x + a2 x2 + · · · =
∞
�
an xn ,
n=0
solve the following equation to obtain an explicit closed-form expression for u(x)
xu� = u + u2 .
Check your solution by substitution.
Hint: You should obtain two solutions, one which is trivial (not a function of x) and another, which
depends on x as well as on an arbitrary parameter.
3. Using D =
d
dx
as a differential operator, and using the following identity
�
�
(D − α)(D − β) f (x)eλx = eλx (D + λ − α)(D + λ − β)f (x)
A. Show that
(D2 + D + 1)
��
� � �
�
Ax2 + Bx + C eix = iAx2 + [iB + (4i + 2)A]x + 2A + (2i + 1)B + iC eix
B. Use you result in A to find the particular solution for y �� + y � + y = xeix
C. From the real and imaginary parts of your result for B, find the particular solution to the following
equations:
(i) y �� + y � + y = x cos(x)
(ii) y �� + y � + y = x sin(x)