Question

Solve $$\frac{dy}{dx}=-\frac{x+y}{3x+3y-4}.$$

Short answer

The equation is solved using a substitution method and separation of variables.

Step-by-step solution

- Recognize the equation form

Identify that the given differential equation $\frac{dy}{dx}=-\frac{x+y}{3x+3y-4}$ is a homogeneous differential equation.

- Substitution

Use the substitution $v=\frac{y}{x}$, which implies $y=vx$. Then, differentiate $y=vx$ to find $\frac{dy}{dx}$.

- Differentiate the substitution

Differentiating $y=vx$, obtain $\frac{dy}{dx}=v+x\frac{dv}{dx}$.

- Substitute and simplify

Substitute $y=vx$ and $\frac{dy}{dx}=v+x\frac{dv}{dx}$ into the original equation to get $v+x\frac{dv}{dx}=-\frac{x+vx}{3x+3vx-4}$.

- Simplify the equation

Simplify the right-hand side: $v+x\frac{dv}{dx}=-\frac{x(1+v)}{x(3+3v)-4}$

- Separate variables

Separate the variables to isolate $v$ and $x$ on different sides: $v+x\frac{dv}{dx}=-\frac{1+v}{3v+3-\frac{4}{x}}$.

- Solve the integral

Integrate both sides with respect to $x$ to find the solution of $v$. Rearrange the equation to facilitate the integration process and perform necessary integrations.

- Back-substitute $v$

After finding $v$ as a function of $x$, back-substitute $v=\frac{y}{x}$ to find $y$ in terms of $x$ and obtain the general solution of the original differential equation.

Key concepts

Differential Equations

Differential equations are mathematical equations that involve an unknown function and its derivatives. They describe how a particular quantity changes with respect to another quantity. In simple terms, they allow us to understand how one variable evolves in relation to another. In this exercise, the differential equation is given as: \ $\frac{dy}{dx}=-\frac{x+y}{3x+3y-4}$ \ This describes how the function $y$ changes with respect to $x$. But to solve this, we need to recognize the type of differential equation we are dealing with and apply appropriate techniques to solve it.

Variable Substitution

Variable substitution is a technique used to simplify differential equations. In this problem, we recognize the equation as homogeneous because the degrees of $x$ and $y$ in the numerator and denominator are the same. We then use substitution to simplify the equation. \ By letting $v=\frac{y}{x}$, we transform $y$ and $\frac{dy}{dx}$ into terms of $x$ and $v$. Given $y=vx$, differentiate it with respect to $x$:\ $\frac{dy}{dx}=v+x\frac{dv}{dx}$. \ Substitute $y$ and $\frac{dy}{dx}$ back into the original equation and simplify. This substitution helps convert a complicated equation into a simpler one that is easier to solve.

Integration

Integration is the process of finding the integral of a function. It is essential in solving differential equations. After substituting variables and simplifying the equation, we use integration to find the solution. \ In this case, after substitution, we separate the variables $v$ and $x$. Our goal is to rearrange the equation in a form where one side involves only $v$ and the other side involves only $x$. \ $v+x\frac{dv}{dx}=-\frac{1+v}{3v+3-\frac{4}{x}}$. \ By integrating both sides with respect to their respective variables, we find the solution in terms of $v$. Once $v$ is solved, we perform the back-substitution $v=\frac{y}{x}$ to find $y$ in terms of $x$, providing us with the general solution to the original differential equation. Integration thus transforms our problem into a solvable form, bridging the substitution step to the final solution.