Here, we show you a step-by-step solved example of exact differential equation. This solution was automatically generated by our smart calculator:
The differential equation $x\cdot dx-y^2dy=0$ 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 ($x$) is equal to zero
Find the derivative of $N(x,y)$ with respect to $x$
The derivative of the constant function ($-y^2$) is equal to zero
Using the test for exactness, we check that the differential equation is exact
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 ($\frac{1}{2}x^2$) is equal to zero
The derivative of $g(y)$ is $g'(y)$
Now take the partial derivative of $\frac{1}{2}x^2$ with respect to $y$ to get
Simplify and isolate $g'(y)$
$x+0=x$, where $x$ is any expression
Rearrange the equation
Set $-y^2$ 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 ($-1$) 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 $2$
Multiplying the fraction by $-1$
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
Multiply the fraction and term in $- \left(\frac{1}{2}\right)x^2$
Multiplying the fraction by $x^2$
Multiply both sides of the equation by $3$
Multiply both sides of the equation by $-1$
We can rename $-3\cdot C_0$ as other constant
Removing the variable's exponent raising both sides of the equation to the power of $\frac{1}{3}$
Find the explicit solution to the differential equation. We need to isolate the variable $y$
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