Final Answer
Step-by-step Solution
Problem to solve:
Specify the solving method
Rewrite the differential equation in the standard form $M(x,y)dx+N(x,y)dy=0$
The differential equation $3y^2dy-2xdx=0$ is exact, since it is written in the standard form $M(x,y)dx+N(x,y)dy=0$, where $M(x,y)$ and $N(x,y)$ are the partial derivatives of a two-variable function $f(x,y)$ and they satisfy the test for exactness: $\displaystyle\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}$. In other words, their second partial derivatives are equal. The general solution of the differential equation is of the form $f(x,y)=C$
Find the derivative of $M(x,y)$ with respect to $y$
The derivative of the constant function ($-2x$) is equal to zero
Find the derivative of $N(x,y)$ with respect to $x$
The derivative of the constant function ($3y^2$) is equal to zero
Using the test for exactness, we check that the differential equation is exact
The integral of a function times a constant ($-2$) is equal to the constant times the integral of the function
Applying the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a number or constant function, in this case $n=1$
Since $y$ is treated as a constant, we add a function of $y$ as constant of integration
Integrate $M(x,y)$ with respect to $x$ to get
The derivative of the constant function ($-x^2$) is equal to zero
The derivative of $g(y)$ is $g'(y)$
Now take the partial derivative of $-x^2$ with respect to $y$ to get
Simplify and isolate $g'(y)$
$x+0=x$, where $x$ is any expression
Rearrange the equation
Set $3y^2$ and $0+g'(y)$ equal to each other and isolate $g'(y)$
Integrate both sides with respect to $y$
The integral of a function times a constant ($3$) is equal to the constant times the integral of the function
Apply the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a number or constant function, such as $2$
Any expression multiplied by $1$ is equal to itself
Find $g(y)$ integrating both sides
We have found our $f(x,y)$ and it equals
Then, the solution to the differential equation is
We need to isolate the dependent variable $y$, we can do that by simultaneously subtracting $-x^2$ from both sides of the equation
Multiply $-1$ times $-1$
Removing the variable's exponent raising both sides of the equation to the power of $\frac{1}{3}$
Divide $1$ by $3$
Simplify $\sqrt[3]{y^{3}}$ using the power of a power property: $\left(a^m\right)^n=a^{m\cdot n}$. In the expression, $m$ equals $3$ and $n$ equals $\frac{1}{3}$
Multiply $3$ times $\frac{1}{3}$
Multiply $3$ times $\frac{1}{3}$
Divide $1$ by $3$
Find the explicit solution to the differential equation. We need to isolate the variable $y$