Question

Question #4daad

Answer 1

Problem 1
D.E.: `-y'''-xy''-2y'=-24x^2`
Is `y=2x^3-6x+10` a solution?

By taking derivatives,

`y'=6x^2-6`, `y''=12x`, `y'''=12`

By plugging the above in the left-hand side of the equation,

`(LHS)=-(12)-x(12x)-2(6x^2-6)=-24x^2`,

which matches the right-hand side of the equation.

Hence, `y=2x^3-6x+10` is a solution.

Problem 2
D.E.: `xy'=y+x^2sinx`
Is `y=-xcosx-x` a solution?

By taking the derivative,

`y'=-cosx+xsinx-1`

By plugging the above in the left-hand side of the equation,

`(LHS)=x(-cosx+xsinx-1)`

`=-xcosx+x^2sinx-x`

by rearranging a bit,

`=(-xcosx-x)+x^2sinx`

by `y=-xcosx-x`,

`=y+x^2sinx`,

which matches the right-hand side of the equation.

Hence, `y=-xcosx-x` is a solution.

I hope that this was helpful.