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 $\left(4x+xy^2\right)dx+\left(y+x^2y\right)dy=0$ 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 a sum of two or more functions is the sum of the derivatives of each function
The derivative of a function multiplied by a constant is equal to the constant times the derivative of the function
The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$
Find the derivative of $N(x,y)$ with respect to $x$
The derivative of a sum of two or more functions is the sum of the derivatives of each function
The derivative of a function multiplied by a constant is equal to the constant times the derivative of the function
The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$
Using the test for exactness, we check that the differential equation is exact
Expand the integral $\int\left(4x+xy^2\right)dx$ into $2$ integrals using the sum rule for integrals, to then solve each integral separately
The integral of a function times a constant ($4$) is equal to the constant times the integral of the function
The integral of a function times a constant ($y^2$) 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 $n=1$
Multiply the fraction and term in $4\cdot \left(\frac{1}{2}\right)x^2$
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 a sum of two or more functions is the sum of the derivatives of each function
The derivative of a function multiplied by a constant is equal to the constant times the derivative of the function
The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$
Multiply the fraction and term in $2\frac{1}{2}x^2y$
Any expression multiplied by $1$ is equal to itself
Divide $2$ by $2$
The derivative of $g(y)$ is $g'(y)$
Now take the partial derivative of $2x^2+\frac{1}{2}y^2x^2$ with respect to $y$ to get
Simplify and isolate $g'(y)$
Rearrange the equation
We need to isolate the dependent variable $g$, we can do that by simultaneously subtracting $x^2y$ from both sides of the equation
Cancel like terms $x^2y$ and $-x^2y$
Set $y+x^2y$ and $x^2y+g'(y)$ equal to each other and isolate $g'(y)$
Integrate both sides with respect to $y$
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 $n=1$
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
Multiplying the fraction by $y^2x^2$
Multiplying the fraction by $y^2$
Combine fractions with common denominator $2$
Factor the polynomial $y^2x^2+y^2$ by it's greatest common factor (GCF): $y^2$
Multiply both sides of the equation by $2$
Simplify $2\left(C_0-2x^2\right)$ using algebra
Multiply $2$ times $-2$
Divide both sides of the equation by $x^2+1$
Removing the variable's exponent
Cancel exponents $2$ and $1$
As in the equation we have the sign $\pm$, this produces two identical equations that differ in the sign of the term $\sqrt{\frac{-4x^2+C_1}{x^2+1}}$. We write and solve both equations, one taking the positive sign, and the other taking the negative sign
The power of a quotient is equal to the quotient of the power of the numerator and denominator: $\displaystyle\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}$
The power of a quotient is equal to the quotient of the power of the numerator and denominator: $\displaystyle\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}$
Multiplying the fraction by $-1$
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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