Stewart Calculus Section 9.1

1. 9.1 Differential Equations
Differential Equation: an equation that
contains an unknown function and one or
more of its derivatives.
Ex: = ( )+

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. Ex: Population over time.
=
=
Goal: find ().

4. Ex: Population over time.
=
=
Goal: find ().
Assumption: the rate of growth of the
population is proportional to the population.
=

5. Ex: Population over time.
=
=
Goal: find ().
Assumption: the rate of growth of the
population is proportional to the population.
=
Solution:
()=

6. Ex: motion of a spring.
=
=
Goal: find ().

7. Ex: motion of a spring.
=
=
Goal: find ().
Hooke’s Law: the force of the spring is
proportional to : =

8. Ex: motion of a spring.
=
=
Goal: find ().
Hooke’s Law: the force of the spring is
proportional to : =
Newton’s Second Law: force equals mass times
acceleration: =

9. Ex: motion of a spring.
=
=
Goal: find ().
Hooke’s Law: the force of the spring is
proportional to : =
Newton’s Second Law: force equals mass times
acceleration: =
Notice that = , we have: =

10. Ex: motion of a spring.
=
=
Goal: find ().
Hooke’s Law: the force of the spring is
proportional to : =
Newton’s Second Law: force equals mass times
acceleration: =
Notice that = , we have: =
Solution:
()= +

11. We often want to find a particular solution
that satisfies a condition of form
( )=
This is called an initial condition. The
overall problem is called an initial-value
problem.

12. We often want to find a particular solution
that satisfies a condition of form
( )=
This is called an initial condition. The
overall problem is called an initial-value
problem.
Ex: = , ( )=

13. We often want to find a particular solution
that satisfies a condition of form
( )=
This is called an initial condition. The
overall problem is called an initial-value
problem.
Ex: = , ( )=
()= ()=

14. We often want to find a particular solution
that satisfies a condition of form
( )=
This is called an initial condition. The
overall problem is called an initial-value
problem.
Ex: = , ( )=
()= ()=
Ex: = , ( )= , ( )=

15. We often want to find a particular solution
that satisfies a condition of form
( )=
This is called an initial condition. The
overall problem is called an initial-value
problem.
Ex: = , ( )=
()= ()=
Ex: = , ( )= , ( )=
()= + ()=