The integrating factor of the differential eq | vedclass

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Question

(D) The given differential equation is $x \frac{dy}{dx} - y = x^3$. Dividing both sides by $x$,we get: $\frac{dy}{dx} - \frac{1}{x} y = x^2$. This is

Vedclass · Accepted Answer

$\frac{1}{x}$

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(D) The given differential equation is $x \frac{dy}{dx} - y = x^3$. Dividing both sides by $x$,we get: $\frac{dy}{dx} - \frac{1}{x} y = x^2$. This is a linear differential equation of the form $\frac{dy}{dx} + P(x)y = Q(x)$,where $P(x) = -\frac{1}{x}$ and $Q(x) = x^2$. The integrating factor $(IF)$ is given by $e^{\int P(x) dx}$. $IF = e^{\int -\frac{1}{x} dx} = e^{-\ln|x|} = e^{\ln|x|^{-1}} = x^{-1} = \frac{1}{x}$. Thus,the correct option is $D$.

The integrating factor of the differential equation $x \frac{dy}{dx} - y = x^3, (x > 0)$ is . . . . . . .

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