Question

Consider the differential equation xy'+ y = x^2 .

Without solving the equation, check which of the following are
solutions:

y(x) = sin(x).

y(x) = (x^2)/3.

y(x) = √x.

y(x) = 5/x.

That is: do not use the method of integrating factors to solve. Just

check by plugging in.

NO SPAM OR RANDOM LINKS!!!​

Answer 1

i have provided the work on a piece of paper.

Answer 2

y = √x is not a solution because y = √x has derivative also involving √x, and rational powers won't get resolved in the ODE. That is,

`y = \sqrt x \implies y' = \frac1{2\sqrt x} \implies xy'+y = \frac x{2\sqrt x}+\sqrt x = \frac32\sqrt x \\eq x^2`

Similarly, y = 5/x is not a solution because its derivative is a rational function that also doesn't get resolved.

`y=\frac5x \implies y=-\frac5{x^2} \implies xy'+y = -\frac5x+\frac5x=0\\eq x^2`

In the same vein, y = sin(x) has derivative y' = cos(x), and these trigonometric expression won't get resolved in this case.

`y=\sin(x)\implies y'=\cos(x) \implies xy'+y=x\cos(x)+\sin(x) \\eq x^2`

So we focus on the remaining candidate:

`y = \frac{x^2}3 \implies y' = \frac{2x}3 \implies xy'+y = \frac{2x^2}3+\frac{x^2}3 = x^2`

and y = x²/3 is the correct choice.