Question

Identify each of the differential equations as type (for example, separable, linear first order, linear second order, etc.), and then solve it. $$3x^3{y}^2{y}^{\prime }-x^2{y}^3=1$$

Short answer

The differential equation is separable, and the general solution is $x{y}^3-yx+\frac{1}{x}=C$.

Step-by-step solution

Identify the Type

The given differential equation is:$$3x^3{y}^2{y}^{\prime }-x^2{y}^3=1$$This equation can be identified as a separable differential equation. Separable differential equations can be written in the form:$$g(y)dy=h(x)dx$$

Rewrite the Equation

Rewrite the given equation to separate the variables:$$3x^3{y}^2{y}^{\prime }=x^2{y}^3+1$$Divide both sides by $y^2(3x^3)$ to isolate $y^{\prime }$:$$y^{\prime }=\frac{x^2{y}^3+1}{3x^3{y}^2}$$Simplify the equation:$$y^{\prime }=\frac{x^{2}y}{3x^3}+\frac{1}{3x^3{y}^2}$$$$y^{\prime }=\frac{y}{3x}+\frac{1}{3x^3{y}^2}$$

Separate the Variables

Separate the variables $y$ and $x$:$$3x{y}^{2}dy=ydx+\frac{dx}{x^2}$$Simplify to get like terms on one side:$$3x{y}^{2}dy-ydx=\frac{dx}{x^2}$$

Integrate Both Sides

Integrate both sides of the equation:$$\int 3x{y}^{2}dy-\int ydx=\int \frac{dx}{x^2}$$For the left side, the integrals are:$$x{y}^3-yx+C$$For the right side, the integral is:$$-\frac{1}{x}$$

Combine and Simplify

Combine and simplify the results of the integrals:$$x{y}^3-yx+C_1=-\frac{1}{x}+C_2$$Combine the constants into one constant, $C$:$$x{y}^3-yx=-\frac{1}{x}+C$$

Solve for the General Solution

Rearrange to get the general solution:$$x{y}^3-yx+\frac{1}{x}=C$$

Key concepts

differential equation types

Differential equations are equations that relate a function with its derivatives. They are widely used to describe various physical phenomena. Here are some common types of differential equations:

• Ordinary Differential Equations (ODEs): These involve functions of a single variable and their derivatives. Examples include first-order and second-order ODEs.
• Partial Differential Equations (PDEs): These involve multiple variables and their partial derivatives.
• Linear Differential Equations: The dependent variable and its derivatives appear linearly. These can be first-order, second-order, etc.
• Non-Linear Differential Equations: The dependent variable or its derivatives appear non-linearly.
• Separable Differential Equations: Can be written as the product of a function of the independent variable and a function of the dependent variable.

In the given problem, we are dealing with a separable differential equation.

separable differential equation

A separable differential equation is one that can be written as the product of a function of the independent variable and a function of the dependent variable. These equations can be separated into two integrals, one involving only the dependent variable and one involving only the independent variable. The general form of a separable differential equation is:
$$\frac{dy}{dx}=g(y)h(x)$$
If we multiply both sides by $dx$, we get:
$$\frac{dy}{g(y)}=h(x)dx$$
Then, we integrate both sides separately to find the general solution. In the exercise, our given differential equation:
$$3x^3{y}^2\frac{dy}{dx}=x^2{y}^3+1$$
is rewritten to separate the variables and then solved by integrating both sides.

solving differential equations step by step

Solving differential equations involves a series of systematic steps:

1. Identify the Type: Determine what type of differential equation you are dealing with (separable, linear, etc.). This step influences the methods used in subsequent steps.
2. Rewrite the Equation: Manipulate the equation to a form that allows for easier handling. For separable differential equations, this involves separating the variables.
3. Separate the Variables: Get all terms involving the dependent variable on one side and all terms involving the independent variable on the other.
4. Integrate Both Sides: Perform integration on both sides of the equation. The integral of each side will yield an expression involving the respective variables.
5. Combine and Simplify: Combine the results from integration and simplify if possible.
6. Solve for the General Solution: Rearrange the simplified equation to isolate the dependent variable, if needed.

In the exercise, these steps are demonstrated clearly.

integrating differential equations

Integration is a crucial step in solving separable differential equations. Integration reverses differentiation, allowing us to find the original function from its derivative. In the context of our problem, we performed the following integrations:

• On the left side, integrate the expression involving $y$
• On the right side, integrate the expression involving $x$

We transformed:
$$\frac{dy}{y^2}=\frac{x^{2}dx}{3x^3}+\frac{dx}{3x^3}$$
We integrated:
$$\begin{array}{rl}\frac{-1}{y}& =-\frac{1}{x}+c\end{array}$$
Combining and simplifying these integrations gave us the general solution for the differential equation:
$$\begin{array}{rl}x{y}^3-yx+\frac{1}{x}& =C\end{array}$$Through careful integration, we derived the final expression.