Which of the given values of x and y make the following pair of matrices equal ? … and …

The differential equation is $\frac{dy}{dx} = x + xy$dxdy​=x+xy.

To find the integrating factor, the equation must be written in standard form:

$$\frac{dy}{dx} + P(x) \cdot y = Q(x)$$dxdy​+P(x)⋅y=Q(x)

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$$\frac{dy}{dx} = x + xy$$dxdy​=x+xy

$$\frac{dy}{dx} = x(1 + y)$$dxdy​=x(1+y)

$$\frac{dy}{dx} - xy = x$$dxdy​−xy=x

The equation is now in standard form with $P(x) = -x$P(x)=−x and $Q(x) = x$Q(x)=x.

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The integrating factor is given by:

$$\text{I.F.} = e^{\int P(x)dx}$$I.F.=e∫P(x)dx

$$\text{I.F.} = e^{\int (-x)dx}$$I.F.=e∫(−x)dx

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$$\int (-x)dx = -\frac{x^2}{2}$$∫(−x)dx=−2x2​

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Therefore, the integrating factor is:

$$\text{I.F.} = e^{-x^2/2}$$I.F.=e−x2/2