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Exam 1 Review 10am Section
Supplemental Instruction
Iowa State University
Tess
Math 267
Basnet
2/9/2016
Exam 1 Information:
Molecular Biology 1414 10-10:50am.
Sections Covered: 1.1-1.3, 2.1-2.5, 3.1-3.
Integral Review
Integral
Solution
Integral
1
∫ 2
𝑑𝑢
𝑢 + 𝑎2
Solution
1
𝑡𝑎𝑛−1 𝑢 + 𝐶
𝑎
∫ 𝑒 𝑢 𝑑𝑢
𝑒𝑢 + 𝐶
1
∫ 𝑑𝑢
𝑢
ln⁡|𝑢| + 𝐶
∫ 𝑐𝑠𝑐𝑢⁡𝑐𝑜𝑡𝑢⁡𝑑𝑢
−cscu + 𝐶
∫ 𝑙𝑛𝑢⁡𝑑𝑢
𝑢⁡𝑙𝑛𝑢 − 𝑢 + 𝐶
∫ 𝑠𝑒𝑐𝑢⁡𝑑𝑢
ln(𝑠𝑒𝑐𝑢 + 𝑡𝑎𝑛𝑢) + 𝐶
∫ 𝑎𝑢 𝑑𝑢
1 𝑢
𝑎 +𝐶
𝑙𝑛𝑎
∫ 𝑠𝑖𝑛𝑢⁡𝑑𝑢
−𝑐𝑜𝑠𝑢 + 𝐶
∫ 𝑠𝑒𝑐 2 𝑢⁡𝑑𝑢
𝑡𝑎𝑛𝑢 + 𝐶
∫
1
√𝑎2 − 𝑢2
𝑑𝑢
∫ 𝑠𝑖𝑛𝑢⁡𝑑𝑢
∫ 𝑢𝑛 𝑑𝑢
𝑢
+𝐶
𝑎
−𝑐𝑜𝑠𝑢 + 𝐶
𝑠𝑖𝑛−1
𝑢𝑛+1
+𝐶
𝑛+1
Additional Methods:
2𝑥
 U-Substitution → find u and du and sub into equation Ex) ∫ 𝑥 2 +1 𝑑𝑥

Integration by Parts → ∫ 𝑢𝑑𝑣 = 𝑢𝑣 − ⁡ ∫ 𝑣𝑑𝑢 Ex) ∫ 𝑥𝑒 𝑥 𝑑𝑥
Method Review
Method
Separable 2.2
Linear (Standard) 2.3
Exact 2.4
Integration Factor 2.4
Bernoulli’s 2.5
General Form
𝑑𝑦
= 𝑔(𝑥)ℎ(𝑦)
𝑑𝑥
𝑑𝑦
+ 𝑃(𝑥)𝑦 = 𝑓(𝑥)
𝑑𝑥
𝑀(𝑥, 𝑦)𝑑𝑥 + 𝑁(𝑥, 𝑦)𝑑𝑦 = 0
𝑀𝑦 − 𝑁𝑥
𝑁𝑥 − 𝑀𝑦
⁡𝑜𝑟⁡
𝑁
𝑀
𝑑𝑦
+ 𝑃(𝑥)𝑦 = 𝑓(𝑥)𝑦 𝑛
𝑑𝑥
Example DE
𝑑𝑦
𝑦𝑒 𝑥
= 𝑒 −𝑦 + 𝑒 −2𝑥−𝑦
𝑑𝑥
𝑑𝑦
(𝑥 + 1)
= 𝑒 𝑥 (1 + 𝑦)
𝑑𝑥
3
𝑑𝑦
3
(1 + + 𝑥)
+𝑦 = −1
𝑦
𝑑𝑥
𝑥
𝑦(𝑥 + 𝑦 + 1)𝑑𝑥 + (𝑥 + 2𝑦)𝑑𝑦 = 0
𝑑𝑦
− 𝑦 = 𝑒 𝑥𝑦2
𝑑𝑥
1060 Hixson-Lied Student Success Center  515-294-6624  sistaff@iastate.edu  http://www.si.iastate.edu
1. Solve the following IVP:
𝑥
𝑑𝑦
+ 𝑦 = 𝑦 2 𝑥 2 𝑙𝑛𝑥; ⁡⁡⁡⁡⁡𝑦(1) = 4
𝑑𝑥
2. Determine if the following DE is exact, if not determine the integrating factor and solve
for general solution:
𝑥 2 𝑦⁡𝑑𝑥 + 𝑦(𝑥 3 + 𝑒 3𝑦 )𝑑𝑦 = 0
3. Consider the following autonomous DE:
a. Determine the critical Points
𝑑𝑃
𝑑𝑥
= −10 + 𝑃2 + 3𝑃
b. Create a phase porrate and classify points as stable, unstable or semi-stable
4. A small metal bar, whose initial tempeature was 20℃, is dropped into a large container
of boiling water.
a. How long will is take the bar to reach 90℃ if it is known that its temperature
increases 2℃ in 1 second?
5. Solve the following DE and give the general solution:
𝑑𝑦
3𝑦𝑥 2
(1 + 3𝑦𝑠𝑒𝑐 2 𝑦 + 4𝑦 4 )
+ 3
= 0⁡
𝑑𝑥 (𝑥 + 2)