MATH 201 - Midterm 1 exam with correct
answers
Ordinary |Differential |Equation |- |verified |answersOne |or |more |dependent |variable |with |respect |to |a |
single |independent |variable
Partial |Differential |Equation |- |verified |answersEquation |involving |partial |derivatives |of |one |or |more |
dependent |variables |of |two |or |more |dependent |variables
Order |of |an |Equation |- |verified |answersThe |order |of |the |highest |derivative |in |the |equation
Linearity |of |an |Equation |- |verified |answersLinear |if |every |y-prime |term |is |independent |of |the |y-
variable; |an |equation |is |NOT |linear |if:
- |a |derivative |coefficient |has |a |y-term |(ex: |y*y')
- |trig |function |of |y |(ex: |sin(y) |)
- |higher |power |than |1 |on |any |y |term |(ex: |y'^2)
Explicit |Solution |- |verified |answersA |solution |where |the |dependent |variable |(y) |is |expressed |solely |in |
terms |of |the |independent |variable |and |constants
Solve |a |First |Order |Initial |Value |Problem |given |an |initial |value |- |verified |answersUse |separation |of |
variables |to |put |the |equation |in |a |form |which |can |be |integrated. |Then, |plug |in |the |initial |value |and |
solve |the |equation.
Interval |definition |of |a |solution |- |verified |answers- |If |we |consider |an |equation |as |a |FUNCTION, |then |
the |domain |is |the |set |of |real |numbers |x |for |which |f(x) |is |defined.
- |If |we |consider |an |equation |as |a |solution |of |a |DIFFERENTIAL |EQUATION, |then |the |interval |of |
definition |I |could |be |taken |to |be |any |interval |over |which |y(x) |is |both |defined |and |differentiable.
- |Finally, |if |we |consider |an |equation |as |a |solution |of |an |INITIAL |VALUE |problem, |then |the |interval |of |
definition |I |is |any |interval |over |which |y(x) |is |defined, |differentiable, |AND |contains |the |initial |point |x=0.
Theorem |1.2.1 |- |Existence |of |a |Unique |Solution |- |verified |answersLet |R |be |a |rectangular |region |in |the |
x-y |plane |that |contains |the |point |x0 |and |y0 |in |its |interior. |If |f(x,y) |and |df/dy |are |continuous |in |R, |then
|there |exists |an |interval |I0 |with |a |width |of |h, |and |on |I0 |there |is |a |unique |function |y(x) |defined |which |
is |a |solution |of |the |initial-value |problem.
Direction |Fields |- |verified |answersGrids |where |the |derivative |of |a |function |is |evaluated |at |various |x |
and |y |values |rather |than |the |function |itself.
Autonomous |First |Order |DE's |- |verified |answersA |function |is |an |autonomous |first-order |DE |if |the |
independent |variable |does |not |appear |explicitly |on |the |right |hand |side |of |the |equation.
answers
Ordinary |Differential |Equation |- |verified |answersOne |or |more |dependent |variable |with |respect |to |a |
single |independent |variable
Partial |Differential |Equation |- |verified |answersEquation |involving |partial |derivatives |of |one |or |more |
dependent |variables |of |two |or |more |dependent |variables
Order |of |an |Equation |- |verified |answersThe |order |of |the |highest |derivative |in |the |equation
Linearity |of |an |Equation |- |verified |answersLinear |if |every |y-prime |term |is |independent |of |the |y-
variable; |an |equation |is |NOT |linear |if:
- |a |derivative |coefficient |has |a |y-term |(ex: |y*y')
- |trig |function |of |y |(ex: |sin(y) |)
- |higher |power |than |1 |on |any |y |term |(ex: |y'^2)
Explicit |Solution |- |verified |answersA |solution |where |the |dependent |variable |(y) |is |expressed |solely |in |
terms |of |the |independent |variable |and |constants
Solve |a |First |Order |Initial |Value |Problem |given |an |initial |value |- |verified |answersUse |separation |of |
variables |to |put |the |equation |in |a |form |which |can |be |integrated. |Then, |plug |in |the |initial |value |and |
solve |the |equation.
Interval |definition |of |a |solution |- |verified |answers- |If |we |consider |an |equation |as |a |FUNCTION, |then |
the |domain |is |the |set |of |real |numbers |x |for |which |f(x) |is |defined.
- |If |we |consider |an |equation |as |a |solution |of |a |DIFFERENTIAL |EQUATION, |then |the |interval |of |
definition |I |could |be |taken |to |be |any |interval |over |which |y(x) |is |both |defined |and |differentiable.
- |Finally, |if |we |consider |an |equation |as |a |solution |of |an |INITIAL |VALUE |problem, |then |the |interval |of |
definition |I |is |any |interval |over |which |y(x) |is |defined, |differentiable, |AND |contains |the |initial |point |x=0.
Theorem |1.2.1 |- |Existence |of |a |Unique |Solution |- |verified |answersLet |R |be |a |rectangular |region |in |the |
x-y |plane |that |contains |the |point |x0 |and |y0 |in |its |interior. |If |f(x,y) |and |df/dy |are |continuous |in |R, |then
|there |exists |an |interval |I0 |with |a |width |of |h, |and |on |I0 |there |is |a |unique |function |y(x) |defined |which |
is |a |solution |of |the |initial-value |problem.
Direction |Fields |- |verified |answersGrids |where |the |derivative |of |a |function |is |evaluated |at |various |x |
and |y |values |rather |than |the |function |itself.
Autonomous |First |Order |DE's |- |verified |answersA |function |is |an |autonomous |first-order |DE |if |the |
independent |variable |does |not |appear |explicitly |on |the |right |hand |side |of |the |equation.
