# Step-by-step Solution

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## Step-by-step explanation

Problem to solve:

$\frac{dy}{dx}\:-\frac{y}{x}=\frac{x}{3y}$

Learn how to solve differential equations problems step by step online.

$\frac{dy}{dx}+\frac{-y}{x}=\frac{x}{3y}$

Learn how to solve differential equations problems step by step online. Solve the differential equation dy/dx-y/x=x/(3y). Multiplying the fraction by -1. We identify that the differential equation \frac{dy}{dx}+\frac{-y}{x}=\frac{x}{3y} is a Bernoulli differential equation since it's of the form \frac{dy}{dx}+P(x)y=Q(x)y^n, where n is any real number different from 0 and 1. To solve this equation, we can apply the following substitution. Let's define a new variable u and set it equal to. Plug in the value of n, which equals -1. Simplify.

## Final Answer

$y=\sqrt{x^{2}\left(\ln\left(\sqrt[3]{x^{2}}\right)+C_0\right)},\:y=-\sqrt{x^{2}\left(\ln\left(\sqrt[3]{x^{2}}\right)+C_0\right)}$
$\frac{dy}{dx}\:-\frac{y}{x}=\frac{x}{3y}$

### Main topic:

Differential equations

### Time to solve it:

~ 0.15 s (SnapXam)