** Final answer to the problem

**

** Step-by-step Solution ** **

** How should I solve this problem?

- Choose an option
- Exact Differential Equation
- Linear Differential Equation
- Separable Differential Equation
- Homogeneous Differential Equation
- Integrate by partial fractions
- Product of Binomials with Common Term
- FOIL Method
- Integrate by substitution
- Integrate by parts
- Load more...

**

**

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

**

**

Isolate the dependent variable $y$

**

**

Differentiate both sides of the equation with respect to the independent variable $x$

**

**

Now, substitute $\frac{dy}{dx}=\frac{1}{2}u^{-\frac{1}{2}}\frac{du}{dx}$ and $y=\sqrt{u}$ on the original differential equation

**

**

Simplify

**

**

We need to cancel the term that is in front of $\frac{du}{dx}$. We can do that by multiplying the whole differential equation by $\frac{1}{2}\sqrt{u}$

**

**

Multiply both sides by $\frac{1}{2}\sqrt{u}$

**

**

Expand and simplify. Now we see that the differential equation looks like a linear differential equation, because we removed the original $y^{-1}$ term

**

**

Divide all terms of the equation by $\frac{1}{4}$

**

**

We can identify that the differential equation has the form: $\frac{dy}{dx} + P(x)\cdot y(x) = Q(x)$, so we can classify it as a linear first order differential equation, where $P(x)=\frac{-1}{\frac{1}{2}x}$ and $Q(x)=\frac{2x}{3}$. In order to solve the differential equation, the first step is to find the integrating factor $\mu(x)$

**

**

To find $\mu(x)$, we first need to calculate $\int P(x)dx$

**

**

So the integrating factor $\mu(x)$ is

**

**

Now, multiply all the terms in the differential equation by the integrating factor $\mu(x)$ and check if we can simplify

**

**

We can recognize that the left side of the differential equation consists of the derivative of the product of $\mu(x)\cdot y(x)$

**

**

Integrate both sides of the differential equation with respect to $dx$

**

**

Simplify the left side of the differential equation

**

**

Applying the property of exponents, $\displaystyle a^{-n}=\frac{1}{a^n}$, where $n$ is a number

**

**

Any expression to the power of $1$ is equal to that same expression

**

**

Solve the integral $\int\frac{2}{3x}dx$ and replace the result in the differential equation

**

**

Replace $u$ with the value $y^{2}$

**

**

Applying the property of exponents, $\displaystyle a^{-n}=\frac{1}{a^n}$, where $n$ is a number

**

**

Multiply the fraction by the term

**

**

Find the explicit solution to the differential equation. We need to isolate the variable $y$

** Final answer to the problem ** **

**