Step-by-step Solution

Solve the differential equation $\frac{dy}{dx}-\left(\frac{y}{x}\right)=\frac{x}{3y}$

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Final Answer

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

Step-by-step Solution

Problem to solve:

$\frac{dy}{dx}\:-\frac{y}{x}=\frac{x}{3y}$
1

Multiplying the fraction by $-1$

$\frac{dy}{dx}+\frac{-y}{x}=\frac{x}{3y}$
2

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

$u=y^{\left(1-n\right)}$
3

Plug in the value of $n$, which equals $-1$

$u=y^{\left(1-1\cdot -1\right)}$
4

Simplify

$u=y^{2}$

Rearrange the equation

$y^{2}=u$

Removing the variable's exponent

$y=\pm \sqrt{u}$

As in the equation we have the sign $\pm$, this produces two identical equations that differ in the sign of the term $\sqrt{u}$. We write and solve both equations, one taking the positive sign, and the other taking the negative sign

$y=\sqrt{u},\:y=-\sqrt{u}$

Let's just take the first equation

$y=\sqrt{u}$
5

Isolate the dependent variable $y$

$y=\sqrt{u}$

The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

$\frac{dy}{dx}=\frac{1}{2}u^{-\frac{1}{2}}$
6

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

$\frac{dy}{dx}=\frac{1}{2}u^{-\frac{1}{2}}\frac{du}{dx}$
7

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

$\frac{1}{2}u^{-\frac{1}{2}}\frac{du}{dx}+\frac{-\sqrt{u}}{x}=\frac{x}{3\sqrt{u}}$
8

Simplify

$\frac{1}{2}u^{-\frac{1}{2}}\frac{du}{dx}+\frac{-\sqrt{u}}{x}=\frac{x}{3\sqrt{u}}$
9

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 $\frac{1}{2}\sqrt{u}$

$\left(\frac{1}{2}u^{-\frac{1}{2}}\frac{du}{dx}+\frac{-\sqrt{u}}{x}=\frac{x}{3\sqrt{u}}\right)\left(\frac{1}{2}\sqrt{u}\right)$
10

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

$\frac{1}{2}\sqrt{u}\left(\frac{1}{2}u^{-\frac{1}{2}}\frac{du}{dx}+\frac{-\sqrt{u}}{x}\right)=\frac{x}{3\sqrt{u}}\frac{1}{2}\sqrt{u}$

Multiplying polynomials $\frac{1}{2}\sqrt{u}$ and $\frac{1}{2}u^{-\frac{1}{2}}\frac{du}{dx}+\frac{-\sqrt{u}}{x}$

$\frac{1}{4}u^{-\frac{1}{2}}\sqrt{u}\left(\frac{du}{dx}\right)+\frac{-\frac{1}{2}u}{x}=\frac{\frac{1}{6}x\sqrt{u}}{\sqrt{u}}$

When multiplying exponents with same base we can add the exponents

$\frac{1}{4}\left(\frac{du}{dx}\right)+\frac{-\frac{1}{2}u}{x}=\frac{\frac{1}{6}x\sqrt{u}}{\sqrt{u}}$

Simplify the fraction $\frac{\frac{1}{6}x\sqrt{u}}{\sqrt{u}}$ by $u$

$\frac{1}{4}\left(\frac{du}{dx}\right)+\frac{-\frac{1}{2}u}{x}=\frac{1}{6}x$
11

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

$\frac{1}{4}\left(\frac{du}{dx}\right)+\frac{-\frac{1}{2}u}{x}=\frac{1}{6}x$
12

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

$\frac{\frac{1}{4}}{\frac{1}{4}}\frac{du}{dx}+\frac{\frac{-\frac{1}{2}u}{x}}{\frac{1}{4}}=\frac{\frac{1}{6}x}{\frac{1}{4}}$

Divide $\frac{1}{4}$ by $\frac{1}{4}$

$1\left(\frac{du}{dx}\right)+\frac{\frac{-\frac{1}{2}u}{x}}{\frac{1}{4}}=\frac{\frac{1}{6}x}{\frac{1}{4}}$

Any expression multiplied by $1$ is equal to itself

$\frac{du}{dx}+\frac{\frac{-\frac{1}{2}u}{x}}{\frac{1}{4}}=\frac{2}{3}x$

Divide fractions $\frac{\frac{-\frac{1}{2}u}{x}}{\frac{1}{4}}$ with Keep, Change, Flip: $\frac{a}{b}\div c=\frac{a}{b}\div\frac{c}{1}=\frac{a}{b}\times\frac{1}{c}=\frac{a}{b\cdot c}$

$\frac{du}{dx}+\frac{-2u}{x}=\frac{2}{3}x$
13

Simplifying

$\frac{du}{dx}+\frac{-2u}{x}=\frac{2}{3}x$
14

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 $P(x)=\frac{-2}{x}$ and $Q(x)=\frac{2}{3}x$. In order to solve the differential equation, the first step is to find the integrating factor $\mu(x)$

$\displaystyle\mu\left(x\right)=e^{\int P(x)dx}$

Compute the integral

$\int\frac{-2}{x}dx$

The integral of the inverse of the lineal function is given by the following formula, $\displaystyle\int\frac{1}{x}dx=\ln(x)$

$-2\ln\left(x\right)$
15

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

$\int P(x)dx=\int\frac{-2}{x}dx=-2\ln\left(x\right)$

Simplify $e^{-2\ln\left(x\right)}$ by applying the properties of exponents and logarithms

$x^{-2}$
16

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

$\mu(x)=x^{-2}$

When multiplying exponents with same base you can add the exponents: $\frac{2}{3}xx^{-2}$

$x^{-2}\frac{du}{dx}+x^{-2}\frac{-2u}{x}=\frac{2}{3}x^{-1}$

Multiplying the fraction by $x^{-2}$

$x^{-2}\frac{du}{dx}+\frac{-2ux^{-2}}{x}=\frac{2}{3}x^{-1}$

Simplify the fraction by $x$

$x^{-2}\frac{du}{dx}-2ux^{-3}=\frac{2}{3}x^{-1}$
17

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

$x^{-2}\frac{du}{dx}-2ux^{-3}=\frac{2}{3}x^{-1}$
18

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

$\frac{d}{dx}\left(x^{-2}u\right)=\frac{2}{3}x^{-1}$
19

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

$\int\frac{d}{dx}\left(x^{-2}u\right)dx=\int\frac{2}{3}x^{-1}dx$
20

Simplify the left side of the differential equation

$x^{-2}u=\int\frac{2}{3}x^{-1}dx$

The integral of a constant by a function is equal to the constant multiplied by the integral of the function

$\frac{2}{3}\int x^{-1}dx$

The integral of the inverse of the lineal function is given by the following formula, $\displaystyle\int\frac{1}{x}dx=\ln(x)$

$\frac{2}{3}\ln\left(x\right)$

Using the power rule of logarithms: $n\log_b(a)=\log_b(a^n)$

$\ln\left(\sqrt[3]{x^{2}}\right)$

Using the power rule of logarithms: $\log_a(x^n)=n\cdot\log_a(x)$

$\frac{2}{3}\ln\left(x\right)$
21

Solve the integral $\int\frac{2}{3}x^{-1}dx$ and replace the result in the differential equation

$x^{-2}u=\frac{2}{3}\ln\left(x\right)$
22

As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$

$x^{-2}u=\frac{2}{3}\ln\left(x\right)+C_0$
23

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

$x^{-2}y^{2}=\frac{2}{3}\ln\left(x\right)+C_0$
24

Using the power rule of logarithms: $n\log_b(a)=\log_b(a^n)$

$x^{-2}y^{2}=\ln\left(\sqrt[3]{x^{2}}\right)+C_0$
25

Using the power rule of logarithms: $\log_a(x^n)=n\cdot\log_a(x)$

$x^{-2}y^{2}=\frac{2}{3}\ln\left(x\right)+C_0$

Divide both sides of the equation by $x^{-2}$

$y^{2}=\frac{\frac{2}{3}\ln\left(x\right)+C_0}{x^{-2}}$

Removing the variable's exponent

$y=\pm \sqrt{\frac{\frac{2}{3}\ln\left(x\right)+C_0}{x^{-2}}}$

The power of a quotient is equal to the quotient of the power of the numerator and denominator: $\displaystyle\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}$

$y=\pm \frac{\sqrt{\frac{2}{3}\ln\left(x\right)+C_0}}{x^{-1}}$

As in the equation we have the sign $\pm$, this produces two identical equations that differ in the sign of the term $\frac{\sqrt{\frac{2}{3}\ln\left(x\right)+C_0}}{x^{-1}}$. We write and solve both equations, one taking the positive sign, and the other taking the negative sign

$y=\frac{\sqrt{\frac{2}{3}\ln\left(x\right)+C_0}}{x^{-1}},\:y=\frac{-\sqrt{\frac{2}{3}\ln\left(x\right)+C_0}}{x^{-1}}$

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

$y=\frac{\sqrt{\frac{2}{3}\ln\left(x\right)+C_0}}{\frac{1}{x}},\:y=\frac{-\sqrt{\frac{2}{3}\ln\left(x\right)+C_0}}{x^{-1}}$

Divide fractions $\frac{\sqrt{\frac{2}{3}\ln\left(x\right)+C_0}}{\frac{1}{x}}$ with Keep, Change, Flip: $a\div \frac{b}{c}=\frac{a}{1}\div\frac{b}{c}=\frac{a}{1}\times\frac{c}{b}=\frac{a\cdot c}{b}$

$y=x\sqrt{\frac{2}{3}\ln\left(x\right)+C_0},\:y=\frac{-\sqrt{\frac{2}{3}\ln\left(x\right)+C_0}}{x^{-1}}$

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

$y=x\sqrt{\frac{2}{3}\ln\left(x\right)+C_0},\:y=\frac{-\sqrt{\frac{2}{3}\ln\left(x\right)+C_0}}{\frac{1}{x}}$

Divide fractions $\frac{-\sqrt{\frac{2}{3}\ln\left(x\right)+C_0}}{\frac{1}{x}}$ with Keep, Change, Flip: $a\div \frac{b}{c}=\frac{a}{1}\div\frac{b}{c}=\frac{a}{1}\times\frac{c}{b}=\frac{a\cdot c}{b}$

$y=x\sqrt{\frac{2}{3}\ln\left(x\right)+C_0},\:y=-x\sqrt{\frac{2}{3}\ln\left(x\right)+C_0}$
26

Find the explicit solution to the differential equation

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

Final Answer

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

Related Formulas:

2. See formulas

Time to solve it:

~ 0.27 s