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 $5x^4dx+20y^{19}dy=0$ 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 ($5x^4$) is equal to zero
Find the derivative of $N(x,y)$ with respect to $x$
The derivative of the constant function ($20y^{19}$) 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 ($5$) 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 $4$
Multiplying the fraction by $5$
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 ($x^{5}$) is equal to zero
The derivative of $g(y)$ is $g'(y)$
Now take the partial derivative of $x^{5}$ with respect to $y$ to get
Simplify and isolate $g'(y)$
$x+0=x$, where $x$ is any expression
Rearrange the equation
Set $20y^{19}$ 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 ($20$) 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 $19$
Multiplying the fraction by $20$
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
Cancel exponents $20$ and $1$
As in the equation we have the sign $\pm$, this produces two identical equations that differ in the sign of the term $\sqrt[20]{C_0-x^{5}}$. We write and solve both equations, one taking the positive sign, and the other taking the negative sign
Combining all solutions, the $2$ solutions of the equation are
Find the explicit solution to the differential equation. We need to isolate the variable $y$
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