Download Calculus 2 Differential Equation Practice Test 1) Consider the

Calculus 2 Differential Equation Practice Test
1) Consider the differential equation
dy
x3
x
– 2y = 2
dx
x +1
(a) Find an integrating factor for this differential equation.
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(b) Solve the differential equation given that y = 1 when x = 1, giving your answer in the form y = f (x).
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2) A curve that passes through the point (1, 2) is defined by the differential equation
dy
= 2x(1+x2 −y).
dx
(a) (i) Use Euler’s method to get an approximate value of y when x = 1.3 , taking steps of 0.1.
Show intermediate steps to four decimal places in a table.
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(ii) How can a more accurate answer be obtained using Euler’s method?
(b) Solve the differential equation giving your answer in the form y = f (x).
dy
+ (2x −1)y = 0 given that y = 2 when x = 0.
dx
A local maximum value of y occurs when x = 0.5. Use Euler’s method with a step value of 0.1 to obtain to this
maximum value approximation of y. Write out your solution in tabular form.
3) Consider the differential equation
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4) Consider the differential equation
(a) Find an integrating factor.
dy
− y cos x = sin x cos x .
dx
€ equation, given that y = − 2 when x = 0 . Give your answer in the form y = f (x).
(b) Solve the differential
y +2
dy
=
and y=1 when x=0, use Euler’s method with interval h = 0.5 to find an approximate
xy +1
dx
value of y when x =1.
5) Given that
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x +3y − 7
dy
6) Show that the linear change of variables X = x − 1, Y = y − 2, transforms the equation
=
to a
3x − y −1
dx
homogeneous form. Hence solve this equation.
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Proof by Induction Practice
n
1) Prove by induction that, for all n ∈ Z + ,
∑ (−1)i i 2 =
i =1
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(−1) n n(n +1)
.
2
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n
2) Find and prove by induction a formula for
1
∑ i(i +1) , where n ∈ Z + .
i =1
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n
3) Find and prove by induction a formula for
∑ (2i −1) , where n ∈ Z + .
i =1
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n
4) Prove by induction that n! > 2 for n ≥ 4.
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5) Prove that for any real number x > -1 and any positive integer n, (1 + x) ≥ 1 + nx