Question

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

Answer 1

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.

Complete step by step answer:
We have to find the general solution to the equation
$\dfrac{{dy}}{{dx}} = \dfrac{{3x}}{y}$
We will use variable separable method where we will separate the terms of a particular variable on each side of the equation. Re-arranging the above differential equation we can write the above equation as:
$\Rightarrow y \cdot dy = 3x \cdot dx$
Now, taking integration to both the sides, we get:

$$\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.

Note: We have used the variable separable method here to solve the question. In the variable separable method we try to separate all the terms of a particular variable on one side of the equation and then integrate both sides to find the solution. The solution of a differential equation is an equation in terms of given variables after eliminating the derivatives.