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# Calculus II - 12

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Stewart Calculus Section 9.1

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• ### Calculus II - 12

1. 1. 9.1 Differential Equations Differential Equation: an equation that contains an unknown function and one or more of its derivatives. Ex: = ( )+
2. 2. 9.1 Differential Equations Differential Equation: an equation that contains an unknown function and one or more of its derivatives. Ex: = ( )+ A function is called a solution if the equations is satisfied. Ex: ( )= C
3. 3. Ex: Population over time. = =Goal: find ().
4. 4. Ex: Population over time. = =Goal: find ().Assumption: the rate of growth of thepopulation is proportional to the population. =
5. 5. Ex: Population over time. = =Goal: find ().Assumption: the rate of growth of thepopulation is proportional to the population. =Solution: ()=
6. 6. Ex: motion of a spring. = =Goal: find ().
7. 7. Ex: motion of a spring. = =Goal: find ().Hooke’s Law: the force of the spring isproportional to : =
8. 8. Ex: motion of a spring. = =Goal: find ().Hooke’s Law: the force of the spring isproportional to : =Newton’s Second Law: force equals mass timesacceleration: =
9. 9. Ex: motion of a spring. = =Goal: find ().Hooke’s Law: the force of the spring isproportional to : =Newton’s Second Law: force equals mass timesacceleration: =Notice that = , we have: =
10. 10. Ex: motion of a spring. = =Goal: find ().Hooke’s Law: the force of the spring isproportional to : =Newton’s Second Law: force equals mass timesacceleration: =Notice that = , we have: =Solution: ()= +
11. 11. We often want to find a particular solutionthat satisfies a condition of form ( )=This is called an initial condition. Theoverall problem is called an initial-valueproblem.
12. 12. We often want to find a particular solutionthat satisfies a condition of form ( )=This is called an initial condition. Theoverall problem is called an initial-valueproblem.Ex: = , ( )=
13. 13. We often want to find a particular solutionthat satisfies a condition of form ( )=This is called an initial condition. Theoverall problem is called an initial-valueproblem.Ex: = , ( )= ()= ()=
14. 14. We often want to find a particular solutionthat satisfies a condition of form ( )=This is called an initial condition. Theoverall problem is called an initial-valueproblem.Ex: = , ( )= ()= ()=Ex: = , ( )= , ( )=
15. 15. We often want to find a particular solutionthat satisfies a condition of form ( )=This is called an initial condition. Theoverall problem is called an initial-valueproblem.Ex: = , ( )= ()= ()=Ex: = , ( )= , ( )= ()= + ()=