For each of the differential equations in find the general soluti… - Vidyadip

Question

For each of the differential equations in find the general solution:
$\text{x}^5\frac{\text{dy}}{\text{dx}}=-\text{y}^5$

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Answer

The given differential equation is
$\text{x}^5\frac{\text{dy}}{\text{dx}}=-\text{y}^5 \ \ \text{or} \ \ \frac{\text{dy}}{\text{dx}}=-\frac{\text{y}^5}{\text{x}^5}$
Separating the variables, we get,
$\frac{1}{\text{y}^5}\text{dy}=-\frac{1}{\text{x}^5}\text{dx}$
Integrating, $\int \frac{\text{1}}{\text{y}^5}\text{dy}=-\int \frac{1}{\text{x}^5}\ \text{dx} \ \text{or}$ $ \int \text{y}^{-5}\text{dy}=-\int\text{x}^{-5}\text{dx}$
$\therefore\ \ \frac{\text{y}^{-4}}{-4}=-\frac{\text{x}^{-4}}{-4}+\text{c}'\ \text{or}$ $\frac{1}{-4\text{y}^4}=\frac{1}{4\text{x}^4}+\text{c}'$
$\text{or} \ \frac{1}{\text{x}^4}+\frac{1}{\text{y}^4}=-4\text{c}'$
$\text{or}\ \text{x}^{-4}+\text{y}^{-4}=\text{c}, \ \text{where c}=-4\text{c}'.$

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