Ordinary Differential Equations
MATH 308H - 523
A. Bonito
February 3
Spring 2015
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Quiz 1
— 5 minute individual quiz;
— Answer the questions in the space provided. If you run out of space, continue
onto the back of the page. Additional space is provided at the end;
— Show and explain all work;
— Underline the answer of each steps;
— The use of books, personal notes, calculator, cellphone, laptop, and communication with others is forbidden;
— By taking this quiz, you agree to follow the university’s code of academic
integrity.
Exercise 1
100%
Find the solution to
ty 0 + 2y = t2 − t + 1,
y(1) =
1
,
2
t > 0.
Ordinary Differential Equations
MATH 308H - 523
A. Bonito
February 3
Spring 2015
Quiz 1: solutions
Exercise 1
100%
First, we rewrite the ODE as
1
2
y0 + y = t − 1 +
t
t
and look for an integrating factor. The latter solves
µ0 =
2
µ,
t
i.e.
ln(µ) = ln(t2 ) + C
for any constant C. Chosing C = 1 and solving for µ yields
µ = t2 .
Hence the original ODE becomes
d 2
(t y) = t3 − t2 + t
dt
or, after integrating from t = 1 and using the inital condition y(1) =
y(t) =
1
1
1 2 1
t − t+ +
.
4
3
2 12t2
1
2