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# How do you find the general solution to $\dfrac{{dy}}{{dx}} = \dfrac{{3x}}{y}$?

Last updated date: 19th Jul 2024
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Hint: The given equation is a differential equation. A differential equation is an equation which involves the derivatives of a variable (which is a dependent variable) with respect to another variable (which is an independent variable).
$\dfrac{{dy}}{{dx}} = f(x)$.
Here, $y$ is the dependent variable
$x$ is the independent variable
and $f(x)$ is a function in terms of the independent variable $x$.
A general solution of ${n^{th}}$ order differential equation can be said to be the solution that includes $n$ arbitrary constants. We can find the general solution of this differential equation by integrating both sides. The general solution of a differential equation is the relation between the x and y variable, that is obtained after the derivatives have been eliminated.

$\dfrac{{dy}}{{dx}} = \dfrac{{3x}}{y}$
$\Rightarrow y \cdot dy = 3x \cdot dx$
$\Rightarrow \int {y \cdot dy} = \int {3x \cdot dx} \\ \Rightarrow \dfrac{{{y^2}}}{2} = \dfrac{{3{x^2}}}{2} + C \\ \Rightarrow \dfrac{{{y^2}}}{2} - \dfrac{{3{x^2}}}{2} = C \\$
Hence, the general solution to $\dfrac{{dy}}{{dx}} = \dfrac{{3x}}{y}$ is given as $\dfrac{{{y^2}}}{2} - \dfrac{{3{x^2}}}{2} = C$, where $C$ is the arbitrary constant.