The solution of r dx + (x - r^2) dr = 0 is | vedclass

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(D) Given the differential equation $r dx + (x - r^2) dr = 0$.Rearranging the terms,we get $r dx = -(x - r^2) dr$.Dividing by $dr$,we have $r \frac{dx

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$rx = \frac{r^3}{3} + c$

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(D) Given the differential equation $r dx + (x - r^2) dr = 0$.Rearranging the terms,we get $r dx = -(x - r^2) dr$.Dividing by $dr$,we have $r \frac{dx}{dr} = r^2 - x$.Rearranging into the standard linear form $\frac{dx}{dr} + P(r)x = Q(r)$,we get $\frac{dx}{dr} + \frac{1}{r}x = r$.The Integrating Factor ($I$.$F$.) is $e^{\int \frac{1}{r} dr} = e^{\log r} = r$.The general solution is given by $x \cdot (I.F.) = \int Q(r) \cdot (I.F.) dr + c$.Substituting the values,$x \cdot r = \int r \cdot r dr + c$.Integrating the right side,$xr = \int r^2 dr + c$.Thus,$xr = \frac{r^3}{3} + c$.

The solution of $r dx + (x - r^2) dr = 0$ is

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