The integrating factor of the differential eq | vedclass

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Question

(A) Given the differential equation: $x \frac{dy}{dx} + 2y = x^2$ Dividing both sides by $x$ (since $x 
eq 0$): $\frac{dy}{dx} + \frac{2}{x} y = x$ T

Vedclass · Accepted Answer

$x^2$

Vedclass · Answer

(A) Given the differential equation: $x \frac{dy}{dx} + 2y = x^2$ Dividing both sides by $x$ (since $x 
eq 0$): $\frac{dy}{dx} + \frac{2}{x} y = x$ This is a linear differential equation of the form $\frac{dy}{dx} + P(x)y = Q(x)$,where $P(x) = \frac{2}{x}$ and $Q(x) = x$. The integrating factor ($I$.$F$.) is given by: $I.F. = e^{\int P(x) dx} = e^{\int \frac{2}{x} dx}$ $= e^{2 \log |x|} = e^{\log |x^2|} = x^2$ Thus,the integrating factor is $x^2$.

The integrating factor of the differential equation $x \cdot \frac{dy}{dx} + 2y = x^2$ is $(x \neq 0)$.

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