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- Equation différentielle exacte
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- Produit de binômes avec terme commun
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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
Apply the formula: $a\frac{dy}{dx}+c=f$$\to \frac{dy}{dx}+\frac{c}{a}=\frac{f}{a}$, where $a=\frac{1}{4}$, $c=\frac{-u}{2x}$ and $f=\frac{x}{6}$
We can identify that the differential equation has the form: $\frac{dy}{dx} + P(x)\cdot y(x) = Q(x)$, so we can classify it as a linear first order differential equation, where $P(x)=\frac{-1}{\frac{1}{2}x}$ and $Q(x)=\frac{2x}{3}$. In order to solve the differential equation, the first step is to find the integrating factor $\mu(x)$
To find $\mu(x)$, we first need to calculate $\int P(x)dx$
So the integrating factor $\mu(x)$ is
Now, multiply all the terms in the differential equation by the integrating factor $\mu(x)$ and check if we can simplify
We can recognize that the left side of the differential equation consists of the derivative of the product of $\mu(x)\cdot y(x)$
Integrate both sides of the differential equation with respect to $dx$
Simplify the left side of the differential equation
Apply the formula: $\frac{x^a}{b}$$=\frac{1}{bx^{-a}}$, where $a=-1$ and $b=3$
Apply the formula: $x^1$$=x$
Solve the integral $\int\frac{2}{3x}dx$ and replace the result in the differential equation
Replace $u$ with the value $y^{2}$
Apply the formula: $x^a$$=\frac{1}{x^{\left|a\right|}}$
Apply the formula: $a\frac{b}{x}$$=\frac{ab}{x}$
Find the explicit solution to the differential equation. We need to isolate the variable $y$
