Match the following differential equation with its solution:

2x^2y" + 3xy' = y

1. y = x^(1/2)

2. y = e^(-4x)

3. y = sin(x)

4. y = 3x + x^(2)

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To match the differential equation with its answer, we need to take each of the functions, differentiate twice and combine to see if the left side of the differential equation matches with the right side.

(1)

`y=x^{1/2}`

` ` `y'=1/2x^{-1/2}`

`y''=-1/4x^{-3/2}`

`LS=-1/2x^{1/2}+3/2x^{1/2}=x^{1/2}=RS`

This is a solution.

(2)

`y=e^{-4x}`

`y'=-4e^{-4x}`

`y''=16e^{-4x}`

`LS=32x^2e^{-4x}-12xe^{-4x}=4x(8x-3)e^{-4x} ne RS`

This is not a solution

(3)

`y=sin x`

`y'=cos x`

`y''=-sin x`

`LS=-2x^2 sinx+3x cos x ne RS`

This is not a solution

(4)

`y=3x+x^2`

`y'=3+2x`

`y''=2`

`LS=4x^2+9x+6x^2=10x^2+9x ne RS`

**The function (1) is a solution to the differential equation.**

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