The integrating factor of x frac{dy}{dx} - 2y | vedclass

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Question

(B) The given differential equation is $x \frac{dy}{dx} - 2y = x^2 + \sin \left( \frac{1}{x^2} \right)$.Divide the entire equation by $x$ to write it

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$\frac{1}{x^2}$

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(B) The given differential equation is $x \frac{dy}{dx} - 2y = x^2 + \sin \left( \frac{1}{x^2} \right)$.Divide the entire equation by $x$ to write it in the standard linear form $\frac{dy}{dx} + P(x)y = Q(x)$:$\frac{dy}{dx} - \frac{2}{x}y = x + \frac{1}{x} \sin \left( \frac{1}{x^2} \right)$.Here,$P(x) = -\frac{2}{x}$.The integrating factor $(IF)$ is given by the formula $IF = e^{\int P(x) dx}$.$IF = e^{\int -\frac{2}{x} dx} = e^{-2 \ln|x|} = e^{\ln|x^{-2}|} = x^{-2} = \frac{1}{x^2}$.

The integrating factor of $x \frac{dy}{dx} - 2y = x^2 + \sin \left( \frac{1}{x^2} \right)$ is

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