Math 2280 Fall 2025 → Participation Worksheet 16
Due December 3, 2025 by 11:59 pm
Name:
ShinYuan
ID: 41498484
The following exercises are a review of Euler’s method
1. Recall that Euler’s method for approximating the solution to the initial value problem y → = f (t, y), y(a) = ω on the interval a ↑ t ↑ b is given by
w0 = ω
wi+1 = wi + hf (ti , wi )
for i = 0, ..., N → 1 where h =
ti = a + ih for i = 0, ..., N .
b→a
, N is the number of subintervals of [a, b], and
N
Use Euler’s method by hand to approximate the solution to the following initialvalue problem:
y → (t) = y → 2t → 3,
y(1) = 5,
1 ↑ t ↑ 2,
h = 0.5.
You may find it easier to work with fractions instead of decimals.
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2. Find the solutions to the following homogeneous equations with constant coe!cients.
(a) y (5) → 7y (4) + 12y →→→ = 0
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(b) y (3) + 3y →→ + 7y → + 5y = 0 (Hint: the characteristic polynomial has exactly one
real, positive rational root)
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3. Given yc = c1 e↑x + c2 xe↑x is the general solution to the equation y →→ + 2y → + y = 0,
use the method of undetermined coe!cients OR variation of parameters to find the
solution of the equation
y →→ + 2y → + y = e↑x .
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4. Solve the separable initial-value problem
↓
dy
y x4 → 16
= 2x3 ,
dx
y(2) = 0
Your solution does not need to be explicitly solved for y.
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5. Solve the first-order linear initial-value problem
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ex
y→ + y = 4 ,
x
x
y(ε) = 3
Solve your solution explicitly for y.
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6. Show that the di”erential equation
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2xy → 2 dx + 3x y + ↓
2 y
3
2 2
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dy = 0
is exact, then solve. You may assume x > 0. Your solution does not need to be
explicitly solved for y.
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