# Match the following differential equation (2x^2)y''+ 3xy" =y with its correct solution: 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 correct solution, we need to take each function, differentiate it twice and put the results in the left side of the equation to see which matches the right side of the equation.

Notice the differential equation simplifies to `x(2x+3)y''=y` .

(1)

`y=x^{1/2}`

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

`y''=-1/4x^{-3/2}` now put in differential equation

`LS=x(2x+3)(-1/4 x^{-3/2})`

`=-1/4(2x+3)x^{-1/2}`

`ne RS`

So (1) is not a solution.

(2)

`y=e^{-4x}`

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

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

`LS=x(2x+3)16e^{-4x} ne RS`

so (2) is not a solution

(3)

`y=sin x`

`y'=cos x`

`y''=-sin x`

`LS=x(2x+3)(-sin x) ne RS`

so (3) is not a solution

(4)

`y=3x+x^2`

`y'=3+2x`

`y''=2x`

`LS=x(2x+3)(2x)`

`=4x^3+6x^2`

`ne RS`

so (4) is not a solution

**This means that none of the given answers are solutions of the differential equation.**