An integrating factor of the differential equ | vedclass

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Question

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

Vedclass · Accepted Answer

$\sqrt{1+x^2}$

Vedclass · Answer

(C) The given differential equation is $(x^2+1) \frac{dy}{dx} + xy = x^3$. Dividing both sides by $(x^2+1)$,we get: $\frac{dy}{dx} + \frac{x}{x^2+1} y = \frac{x^3}{x^2+1}$. This is a linear differential equation of the form $\frac{dy}{dx} + Py = Q$,where $P = \frac{x}{x^2+1}$. The integrating factor $(IF)$ is given by $e^{\int P dx}$. $IF = e^{\int \frac{x}{x^2+1} dx}$. Let $u = x^2+1$,then $du = 2x dx$,so $x dx = \frac{1}{2} du$. $IF = e^{\frac{1}{2} \int \frac{1}{u} du} = e^{\frac{1}{2} \ln(x^2+1)} = e^{\ln((x^2+1)^{1/2})} = \sqrt{x^2+1}$. Thus,the correct option is $C$.

An integrating factor of the differential equation $(x^2+1) \frac{dy}{dx} + xy = x^3$ is

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