dx%2Fdy%2Bx%5E3y%5E2%3D0.jpeg "(x^2y^3-1/1+9x^2)dx/dy+x^3y^2=0")
Differential Equation: Solving $(x^2y^3-\frac{1}{1+9x^2})dx/dy+x^3y^2=0$
Introduction
Differential equations are a fundamental concept in mathematics and are used to model various phenomena in fields such as physics, engineering, and economics. In this article, we will focus on solving a particular differential equation: $(x^2y^3-\frac{1}{1+9x^2})dx/dy+x^3y^2=0$.
The Differential Equation
The given differential equation is:
$(x^2y^3-\frac{1}{1+9x^2})\frac{dx}{dy}+x^3y^2=0$
This is a first-order ordinary differential equation (ODE), where $x$ is the independent variable and $y$ is the dependent variable.
Separation of Variables
To solve this differential equation, we can use the separation of variables method. This involves separating the variables $x$ and $y$ and integrating both sides of the equation separately.
Let's start by multiplying both sides of the equation by $dy$ to get:
$(x^2y^3-\frac{1}{1+9x^2})dx+x^3y^2dy=0$
Now, we can separate the variables by rearranging the equation as follows:
$\int\frac{dx}{x^2y^3-\frac{1}{1+9x^2}}=\int -x^3y^2dy$
Solving the Integral
To solve the integral, we need to integrate both sides of the equation. Let's start with the left-hand side:
$\int\frac{dx}{x^2y^3-\frac{1}{1+9x^2}}=\int\frac{1}{x^2y^3-\frac{1}{1+9x^2}}\cdot\frac{dx}{dx}$
To simplify the integral, we can use the substitution $u=x^2y^3-\frac{1}{1+9x^2}$, which implies $du=(2xy^3-\frac{18x}{(1+9x^2)^2})dx$. Then, the integral becomes:
$\int\frac{du}{u}=\ln|u|+C_1$
Substituting back, we get:
$\ln|x^2y^3-\frac{1}{1+9x^2}|=C_1$
Now, let's move on to the right-hand side of the equation:
$\int -x^3y^2dy=-\int x^3y^2dy$
This integral can be evaluated using integration by parts or by using a substitution. After some manipulation, we get:
$\frac{1}{3}x^3y^3=C_2$
Combining the Results
Combining the results from both integrals, we get:
$\ln|x^2y^3-\frac{1}{1+9x^2}|=-\frac{1}{3}x^3y^3+C$
where $C=C_1+C_2$ is the constant of integration.
This is the general solution to the differential equation $(x^2y^3-\frac{1}{1+9x^2})dx/dy+x^3y^2=0$.
Conclusion
In this article, we have solved the differential equation $(x^2y^3-\frac{1}{1+9x^2})dx/dy+x^3y^2=0$ using the separation of variables method. The general solution to the equation is given by $\ln|x^2y^3-\frac{1}{1+9x^2}|=-\frac{1}{3}x^3y^3+C$, where $C$ is the constant of integration. This solution can be used to model various real-world phenomena that can be described by this differential equation.
