Solved example of exact differential equation
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
Group the terms of the equation
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 $y$
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