Solved example of first order differential equations
Rewrite the differential equation in the standard form $M(x,y)dx+N(x,y)dy=0$
The differential equation $4ydy-5x^2dx=0$ 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 ($-5x^2$) is equal to zero
Find the derivative of $N(x,y)$ with respect to $x$
The derivative of the constant function ($4y$) 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 $2$
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 ($\frac{-5x^{3}}{3}$) is equal to zero
The derivative of $g(y)$ is $g'(y)$
Now take the partial derivative of $\frac{-5x^{3}}{3}$ with respect to $y$ to get
Simplify and isolate $g'(y)$
$x+0=x$, where $x$ is any expression
Rearrange the equation
Set $4y$ 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 ($4$) 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 $n=1$
Find $g(y)$ integrating both sides
We have found our $f(x,y)$ and it equals
Then, the solution to the differential equation is
Combine all terms into a single fraction with $3$ as common denominator
Multiply $2$ times $3$
Multiply both sides of the equation by $3$
We can rename $3C_0$ as other constant
We need to isolate the dependent variable $y$, we can do that by subtracting $-5x^{3}$ from both sides of the equation
Divide both sides of the equation by $6$
Removing the variable's exponent
As in the equation we have the sign $\pm$, this produces two identical equations that differ in the sign of the term $\sqrt{\frac{5x^{3}+C_0}{6}}$. We write and solve both equations, one taking the positive sign, and the other taking the negative sign
Find the explicit solution to the differential equation
Access detailed step by step solutions to thousands of problems, growing every day!