Math 257/316 Assignment 1, 2026 Summer Term1
Due Monday May 25 Submit online in a PDF document on
Canvas by 11:59 pm of the due date
Problem 1 (Submit): (ODE Review) Find the general solutions of
the following equations:
a. y ′′ − 2y ′ + 2y = ex cos x
b. (1 − x)y ′ = y ln y
c. 1 + x2 y ′ + 2xy = 1
d. 2x2 y ′′ − xy ′ + y = 0
e. 2x2 y ′′ − xy ′ + y = x1/2
Problem 2 (Do not Submit): (Power series solution): Consider
the following linear ODEs:
y ′ + (1 + x)y = 0
′′
y + 4y = 0
(1)
(2)
a. Use the methods you learned for solving ODEs to solve equations (1)
and (2).
b. Expand the solutions to (1) and (2) as Taylor series about the point
x0 = 0.
c. Now for each of equations (1) and (2) assume a power series solution
of the form
∞
X
y(x) =
an xn
(3)
n=0
obtain a recursion for the coefficients an . Use these recursions to determine
the series representation of each solution. Compare these results to the series obtained in part b above.
Problem 3 (Submit): (Power series solution): Consider the ODE:
(1 + x2 )y ′′ + 4xy ′ + 2y = 0.
(4)
(a) Compute the first 3 nonzero terms of power series expansion about
x = 0 for two linearly independent solutions.
(b) Use the ratio test to determine the radius of convergence of the series.
Could your result have been predicted by inspection?
Problem 4 (Do not submit): (Power series solution): Consider
the following first order linear ODEs:
1
y ′ + (1 − 2x)y = 0
′
xy + (2 − x)y = 0
(5)
(6)
a. Solve the differential equations (5) and (6) using the appropriate
integrating factors.
b. Expand the solution to (5) as Taylor series about the point x0 = 0.
Expand the exponential in the solution to (6) as a power series.
c. Now for (5) assume a power series solution of the form
y(x) =
∞
X
an xn
(7)
n=0
obtain a recursion for the coefficients an . Use these recursions to determine
the series representation of the solution. Compare this result to the series
obtained in part b above.
d. Try using the same power series expansion (7) to solve (6). What
happens?
e. Consider the following recursive strategy to generate an approximate
solution to (6). Rewrite (6) as
xy ′ + 2y = xy
(8)
Now assuming x → 0 and discarding the right hand side of (8), find a first
order approximation y0 as the solution to
xy0′ + 2y0 = 0
Now substitute y0 on the right side of (8) and solve for y1
xy1′ + 2y1 = xy0
Continue this process till you obtain y2 . How does y2 compare with the
series solution to (6) obtained in b? Can you use this series to motivate a
modification to the series expansion (7) that would be appropriate to use to
obtain a series solution to (6)?
Problem 5 (Do not submit): (Power series solution): Compute
the first 3 nonzero terms of the power series expansion about x = 0 of two
linearly independent solutions of the ODE: y ′′ − (sinx)y = 0.
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Problem 6 (Do not submit)– Frobenius series: Consider the following Gauss’s hypergeometric equation:
x(1 − x)y ′′ + (c − (a + b + 1)x)y ′ − aby = 0.
(9)
(a) Classify the points 0 < x < ∞ as ordinary points, regular singular
points, or irregular singular points.
(b) Find the exponents at the regular singular point(s)
(c) Assuming that 1 − c is not a positive integer, find the series solution
of (9), in the neighborhood of x = 0. What would you expect the radius of
convergence of this series to be?
(d) Assuming that 1−c is not an integer or zero, find the second solution
for 0 < x < 1.
Problem 6 (Do not submit)–Singular Points at Infinity: (a) Consider the equation
4
2
y ′′ + y ′ + 2 y = 0.
x
x
Use the change of variable t = x1 to identify and classify singular points
at x =P
∞. Compute the indices of the associated Frobenius series (i.e. if
y = xr an xn , determine r).
(b) Verify that
c−x ′ a
y ′′ +
y − y=0
x
x
has an irregular singular point at x = ∞.
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