Problem set 4
1. In the following equations classify x = 0 as ordinary, regular singular, irregular
singular.
(a) x2 (1 + x2 )y′′ + xy = 0
(b) xy′′ + x5 y′ + y = 0
(c) x2 y′′ − y′ + xy = 0
(d) (x2 − x)y′′ − 3xy′ + (1 − cos x)y = 0
(e) (x2 − 1)x2 y′′ + y′ + y = 0
x2 − 4 ′
(f) (x2 − 1)y′′ + 2
y + xy = 0
x
(g) (cos x)y′′ − (sin x)y = 0
(h) x4 y′′ − (x2 sin x)y′ + 2x2 y = 0
(i) x2 y′′′ + (1 + x)y′ = 0
(j) y′′ + P(x)y′ + Q(x)y = 0, having solutions y1 = x, and y2 = x2 .
2. Without solving the ODE, find a lower bound for the radius of convergence of
the power series solution to the second order ODE
x 1
(x3 + x2 + x + 1)y′ + y′
+ y=0
x2 − 2x + 4 x−3
a) near the ordinary point x = 0, b) near the ordinary point x = 1.
3. Check that 0 is an ordinary point to the following ODE then solve them in a
neighborhood of x = 0. (It is enough to write the first four non zero terms of the
series of each LI solution.)
a) y′′ − xy = 0, (Airy equation).
b) y′′ − 2xy′ + 2y = 0, (Hermite equation of order 1)
c) (1 − x2 )y′′ − 2xy′ + 2y = 0, |x| < 1. (Legendre equation of order 1)
d) (1 − x )y − xy + 4y = 0,
2 ′′ ′
|x| < 1. (Chebyshev equation of order 2)
4. Solve the following IVP using a power series centered at 0. (It is enough to
write the first five non zero terms of the series)
a) y′′ + (x + 3)y′ − 2y = 0, y(0) = 1, y′ (0) = 2.
b) (x2 − 5x + 6)y′′ − 5y′ − 2y = 0, y(0) = 1, y′ (0) = 1.
c) y′′ + (cos x)y = 0, y(0) = 2, y′ (0) = 6.
d) y′′ − e7x y′ + xy = 0, y(0) = 2, y′ (0) = 1.
1
2
5. Bessel Equation. Bessel equation of order α > 0 has the form
x2 y′′ + xy′ + (x2 − α2 )y = 0, x > 0.
a) Show that 0 is a regular singular point to the Bessel ODE.
b) Show that if y = xr ∞ n
P
n=0 cn x is a solution to the Bessel equation then replacing
in the ODE we get
h i X∞ nh i o
c0 r − α + c1 (r + 1) − α x +
2 2 2 2
(k + r)2 − α2 ck + ck−2 xk = 0,
k=2
c) We now solve the Bessel equation for some choices of α. Let r1 = α and r2 = −α
the solutions to the indicial equation.
i) For α = 14 , find the general solution to the corresponding Bessel equation.
It is enough to write the first three non zero terms of the series of each
linearly independent solution.
3
ii) For α = , find the general solution to the corresponding Bessel equation.
2
It is enough to write the first three non zero terms of the series of each
linearly independent solution.
1
iii) For α = . Use only the smaller index r2 to find the general solution to the
2
ODE in this case. Express your solution in terms of elementary functions.
iv) For α = 1. Show that the indices r1 and r2 give linearly dependent solution.
Let y1 be the solution obtained using the indices. Find a second solution
y2 that is linearly independent to y1 . Then conclude the general solution
in this case. Hint: you may need to use long division to compute y2 .
v) For α = 0, show that in this case r1 = r2 , then find a solution y1 . Deduce
a second solution y2 that is linearly independent to y1 . Then conclude
the general solution in this case. Hint: you may need to use long division to
compute y2 .
6. Use Frobenius method to find one solution to the following ODE. Express
your obtained solution in terms of elementary functions then deduce the general
solution. We assume in all the ODEs that we are solving for x > 0.
a) xy′′ + y′ − (x + 1)y = 0.
b) xy′′ + (1 − x)y′ − y = 0. (Laguerre equation of order −1)
c) (x2 + 2x)y′′ − 2(x2 +
2x − 1)y + (x + 2x − 2)y = 0.
′ 2
2
d) xy′′ + xy′ − 2 + y = 0.
x
e) xy′′ + (1 − 4x2 )y′ − (4x − 4x3 )y = 0.
3
7. Consider the ODE for x > 0
x4 y′′ + 4y = 0. (⋆)
(a) Show that x = 0 is an irregular singular point.
(b) Let u(t) = y(1/t). Show that u solves
2
u′′ + u′ + 4u = 0. (⋆⋆)
t
(c) Show that t = 0 is a regular singular point to (⋆⋆). Find u using Frobenius
method. Express u in terms of elementary functions, then conclude the general
solution y to (⋆).
8. (Boyce, DiPrima). Consider the differential equation
x3 y′′ + αxy′ + βy = 0,
with α, β real constants and α , 0.
a) Show that x = 0 is an irregular singular point.
b) By attempting to determine a solution of the form xr ∞ n
P
n=0 cn x , show that the
indicial equation for r is linear and consequently there is only one formal
solution of the assumed form.
c) Show that if β/α is an integer larger than −1 that is β/α = −1, 0, 1, 2, · · · then the
formal series terminates and therefore is an actual solution. For other values
of β/α show that the formal series solution has a zero radius of convergence
and so does not represent an actual solution in any interval.
d) Using the parts above, find the general solution to the follow ODEs:
i) x3 − 4xy′ + 4y = 0.
ii) x3 y′′ + 4xy′ + 4y = 0.
