smc-7296-70-IVP Terminology

Calculus, EXT1 C3 2022 SPEC1 2

Solve the initial value problem `(dy)/(dx) = -x sqrt(4-y^2)` given that `y(2) = 0`. Give your answer in the form `y = f(x)`. (3 marks)

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

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`int(dy)/(sqrt(4-y^(2)))`	`=int-x\ dx`
`sin^(-1)((y)/(2))`	`=-(1)/(2)x^(2)+c`

`y(2)=0\ \=> \ c=2`

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

Find the particular solution to the initial value problem `(dy)/(dx)=e^(2x+3y)` that passes through the point `(0,0)`. (3 marks)

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`y=-1/3ln((5-3e^(2x))/2)`

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`text{Passes through}\ (0,0):`

`-1/3e^0=1/2e^0+c\ \ =>\ \ c=5/6`

Find the particular solution to the initial value problem `(dy)/(dx)=(2y+1)(x-3)` that passes through the point `(2,-1)`. (4 marks)

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`y=-1/2(e^((x-2)(x-4))+1)`

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`text{Passes through}\ (2,-1):`

`1/2ln|-1|=2-6+c\ \ =>\ \ c=4`

Find an expression for `y` in terms of `x` given the initial value problem

`dy/dx=4y-3` and when `x=-2, \ y=1`. (3 marks)

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`y=(e^(4(x+2))+3)/4`

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`text{When}\ \ y=1, x=-2:`

`1/4ln(4-3)=-2+c\ \ =>\ \ c=2`