We could not solve this problem by using the method: Exact Differential Equation
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
$u=y^{\left(1-n\right)}$
2
Plug in the value of $n$, which equals $-1$
$u=y^{\left(1-1\cdot -1\right)}$
3
Simplify
$u=y^{2}$
Intermediate steps
4
Isolate the dependent variable $y$
$y=\sqrt{u}$
Intermediate steps
5
Differentiate both sides of the equation with respect to the independent variable $x$
Expand and simplify. Now we see that the differential equation looks like a linear differential equation, because we removed the original $y^{-1}$ term
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 $Q(x)=\frac{\frac{1}{6}x}{\frac{1}{4}}$. In order to solve the differential equation, the first step is to find the integrating factor $\mu(x)$
$\displaystyle\mu\left(x\right)=e^{\int P(x)dx}$
Intermediate steps
14
To find $\mu(x)$, we first need to calculate $\int P(x)dx$
Solving a math problem using different methods is important because it enhances understanding, encourages critical thinking, allows for multiple solutions, and develops problem-solving strategies. Read more