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: = , ( )= , ( )= ()= + ()=

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