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2.2 Exercises Applications
Math 3331 Differential Equations2.2 Solutions to Separable Equations
Jiwen He
Department of Mathematics, University of Houston
[email protected]/∼jiwenhe/math3331
Jiwen He, University of Houston Math 3331 Differential Equations Summer, 2014 1 / 21
2.2 Exercises Applications General Method Implicitly Solutions
2.2 Solutions to Separable Equations
General Method: Separation of Variables
Separable EquationExponential EquationGeneral MethodExplicit SolutionImplicitly Defined Solutions
Applications
Radioactive Decay and Half-lifeNewton’s Law of Cooling
Worked out Examples from Exercises:
Find General Solutions: 1, 3, 5, 9, 11Find Solutions to IVPs and IoEs: 13, 15, 17, 19Application: 26, 33
Jiwen He, University of Houston Math 3331 Differential Equations Summer, 2014 2 / 21
2.2 Exercises Applications General Method Implicitly Solutions
The General Method: Separation of Variables
Jiwen He, University of Houston Math 3331 Differential Equations Summer, 2014 3 / 21
2.2 Exercises Applications General Method Implicitly Solutions
Example
Jiwen He, University of Houston Math 3331 Differential Equations Summer, 2014 4 / 21
2.2 Exercises Applications General Method Implicitly Solutions
Example: Exponential Equation
Jiwen He, University of Houston Math 3331 Differential Equations Summer, 2014 5 / 21
2.2 Exercises Applications General Method Implicitly Solutions
Example: IVP
Jiwen He, University of Houston Math 3331 Differential Equations Summer, 2014 6 / 21
2.2 Exercises Applications General Method Implicitly Solutions
Implicitly Defined Solutions
Find sols. of x ′ = 2tx/(1 + x)s.t. x(0) = 1, x(0) = −2, andx(0) = 0.
Answer:
(1 + 1/x)dx = 2tdt, x 6= 0
⇒ x + ln(|x |) = t2 + C
(i) For x(0) = 1
⇒ C = 1
x + ln x − 1 = t2
⇒ x(t) implicitly defined.
Jiwen He, University of Houston Math 3331 Differential Equations Summer, 2014 7 / 21
2.2 Exercises Applications
Exercise 2.2.1
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2.2 Exercises Applications
Exercise 2.2.3
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2.2 Exercises Applications
Exercise 2.2.5
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2.2 Exercises Applications
Exercise 2.2.9
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2.2 Exercises Applications
Exercise 2.2.11
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2.2 Exercises Applications
Exercise 2.2.13
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2.2 Exercises Applications
Exercise 2.2.15
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2.2 Exercises Applications
Exercise 2.2.17
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2.2 Exercises Applications
Exercise 2.2.19
Jiwen He, University of Houston Math 3331 Differential Equations Summer, 2014 16 / 21
2.2 Exercises Applications Radioactive Decay Newton’s Law of Cooling
Radioactive Decay
Jiwen He, University of Houston Math 3331 Differential Equations Summer, 2014 17 / 21
2.2 Exercises Applications Radioactive Decay Newton’s Law of Cooling
Exercise 2.2.25
Jiwen He, University of Houston Math 3331 Differential Equations Summer, 2014 18 / 21
2.2 Exercises Applications Radioactive Decay Newton’s Law of Cooling
Exercise 2.2.26
Jiwen He, University of Houston Math 3331 Differential Equations Summer, 2014 19 / 21
2.2 Exercises Applications Radioactive Decay Newton’s Law of Cooling
Newton’s Law of Cooling
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2.2 Exercises Applications Radioactive Decay Newton’s Law of Cooling
Exercise 2.2.33
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