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Calculus, EXT1 C3 2016 SPEC1 10

Solve the differential equation  `sqrt(2-x^2) (dy)/(dx) = 1/(2-y)`, given that  `y(1) = 0`. Express `y` as a function of  `x`.  (4 marks)

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`y = 2-sqrt(4 + pi/2-2 sin^(-1)(x/sqrt 2))`

Show Worked Solution
`sqrt(2-x^2) *(dy)/(dx)` `= 1/(2-y)`
`(2-y)* (dy)/(dx)` `= 1/sqrt(2-x^2)`
`int 2-y\ dy` `= int 1/(sqrt(2-x^2))\ dx`
`2y-y^2/2` `= sin^(-1) (x/sqrt 2) + c`

 
`text(Given)\ \ y(1) = 0:`

♦ Mean mark 46%.

`0=sin^(-1) (1/sqrt 2) + c`

`c=-pi/4`

`2y-y^2/2` `= sin^(-1) (x/sqrt 2)-pi/4`
`y^2-4y` `= -2 sin^(-1) (x/sqrt 2) + pi/2`
`(y-2)^2-4` `= -2 sin^(-1) (x/sqrt 2) + pi/2`
`(y-2)^2` `= 4 + pi/2-2 sin^(-1) (x/sqrt 2)`
`(y-2)` `= +- sqrt(4 + pi/2-2 sin^(-1) (x/sqrt 2))`
`y` `=2 +- sqrt(4 + pi/2-2 sin^(-1) (x/sqrt 2))`

 
`text(Given)\ \ y=0\ \ text(when)\ \ x=1:`

`:. y=2-sqrt(4 + pi/2-2 sin^(-1) (x/sqrt 2))`

Calculus, EXT1 C3 2017 SPEC2-N 10 MC

A solution to the differential equation  `(dy)/(dx) = (cos(x + y) - cos(x - y))/(e^(x + y))`  can be obtained from

  1. `int e^y/(sin(y))\ dy = -int (2 sin(x))/e^x\ dx`
  2. `int e^y/(cos(y))\ dy = int 2/e^x\ dx`
  3. `int e^y/(cos(y))\ dy = -int (2 cos(x))/e^x\ dx`
  4. `int e^y/(cos(y))\ dy = int (2 sin(x))/e^x\ dx`
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`A`

Show Worked Solution
`dy/dx` `=(cos(x + y) – cos(x – y))/(e^(x + y))`
`(dy)/(dx)` `= (cos(x) cos(y) – sin(x) sin(y) – cos(x) cos(y) – sin(x) sin(y))/(e^x ⋅ e^y)`
`e^y *(dy)/(dx)` `= (-2 sin(x) sin(y))/(e^x)`
`e^y/(sin(y)) *(dy)/(dx)` `= (-2 sin(x))/(e^x)`
`:. int e^y/(sin(y))\ dy` `= -int (2 sin(x))/e^x\ dx`

 
`=>   A`

Calculus, EXT1 C3 2018 SPEC2 9 MC

A solution to the differential equation  `(dy)/(dx) = 2/{sin(x + y) - sin(x - y)}`  can be obtained from

  1. `int 1\ dx = int 2 sin(y)\ dy`
  2. `int cos(y)\ dy = int text{cosec}(x)\ dx`
  3. `int cos(x)\ dx = int text{cosec}(y)\ dy`
  4. `int sec(x)\ dx = int sin(y)\ dy`
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`D`

Show Worked Solution
`(dy)/(dx)` `= 2/{sin(x) cos(y) + sin(y) cos(x) – (sin(x) cos(y) – sin(y) cos(x))}`
  `= 2/{2 sin(y) cos(x)}`
  `= 1/{sin(y) cos(x)}`

 
`sin(y) *(dy)/(dx)= sec(x)`

`int sin (y)\ dy= int sec(x)\ dx`

`=>  D`