The integrating factor of the differential eq | vedclass

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Question

(C) Given the differential equation: $\frac{dy}{dx}(1+x) - xy = 1-x$. Divide the entire equation by $(1+x)$ to write it in the standard linear form $\

Vedclass · Accepted Answer

$(1+x)e^{-x}$

Vedclass · Answer

(C) Given the differential equation: $\frac{dy}{dx}(1+x) - xy = 1-x$. Divide the entire equation by $(1+x)$ to write it in the standard linear form $\frac{dy}{dx} + P(x)y = Q(x)$: $\frac{dy}{dx} - \frac{x}{1+x}y = \frac{1-x}{1+x}$. Here,$P(x) = -\frac{x}{1+x}$. The integrating factor $(IF)$ is given by $e^{\int P(x) dx}$: $IF = e^{\int -\frac{x}{1+x} dx} = e^{-\int \frac{x+1-1}{1+x} dx} = e^{-\int (1 - \frac{1}{1+x}) dx}$. $IF = e^{-(x - \ln|1+x|)} = e^{-x + \ln|1+x|} = e^{-x} \cdot e^{\ln|1+x|}$. Since $e^{\ln|1+x|} = 1+x$,we get $IF = (1+x)e^{-x}$. Thus,the correct option is $C$.

The integrating factor of the differential equation $\frac{dy}{dx}(1+x) - xy = 1-x$ is . . . . . . .

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