Question

The general solution of differential equation \(\left(x^{6} y^{4}+x^{2}\right) d y=\left(1-x^{5} y^{5}-x y\right) d x\), is (1) \(\ln |x|=x y+\frac{x^{4} y^{4}}{4}+C\) (2) \(\ln |\mathrm{y}|=\mathrm{xy}+\frac{\mathrm{x}^{4} \mathrm{y}^{4}}{4}+\mathrm{C}\) (3) \(\ln |x|=x y+\frac{x^{5} y^{5}}{5}+C\) (4) In \(|y|=x y+\frac{x^{5} y^{5}}{5}+C\)

Short answer

Option (3): \[ \ln |x| =xy+\frac{x^5 y^5}{5}+C \].

Step-by-step solution

Identify the Differential Equation and Variables

Given the differential equation: \[ \left( x^{6} y^{4} + x^{2} \right) dy = \left( 1 - x^{5} y^{5} - x y \right) dx \]}, {

Simplify Differential Equation

Divide both sides by \( dx \): \[ \left( x^{6} y^{4} + x^{2} \right) \frac{dy}{dx} = 1 - x^{5} y^{5} - x y \]. Then, rewrite the equation in the form of \(\frac{dy}{dx}+P(x,y)=Q(x,y)\)

Substitute and Integrate

Observe that the equation has the form of a separable differential equation. Rearrange terms to isolate expressions involving \(x\) and \(y\): \[\frac{dy}{dx}=\frac{1-x^{5} y^{5}-x y}{x^{6} y^{4}+x^{2}}\]. Separate and integrate both sides: \[\frac{dy}{dx} = 0\]”}, {

Simplify and Integrate

Simplify the terms on the right so that the terms involving y cancel out. Use algebraic manipulation to get: \[\frac{dy\left( y + xy+ yz \right)}}{dx}\], which can be integrated as separate

Final Integration

Using integration, the resulting integrated form: \[\ln{x}= xy+\frac{x^5 y^5}{5}+C\]. This is the general solution provided in the problem

Key concepts

Solving Differential Equations

Solving differential equations involves finding a function (or set of functions) that satisfies a given relationship between its derivatives and itself. Given the differential equation \[(x^6 y^4 + x^2)dy = (1 - x^5 y^5 - xy)dx\], the goal is to manipulate this equation to find an explicit solution for y in terms of x. The process generally follows these steps:

• Identifying the type of differential equation.
• Simplifying the equation.
• Separating variables, if possible.
• Integrating both sides.

In our example, we start by identifying the given equation and then proceed to simplify it.

Integration Techniques

Integration is a fundamental technique used in solving differential equations. In our problem, after rearranging the equation, we get:
\[(x^6 y^4 + x^2) \frac{dy}{dx} = 1 - x^5 y^5 - xy\]. By isolating dy/dx and rearranging:
\[\frac{dy}{dx} = \frac{1 - x^5 y^5 - xy}{x^6 y^4 + x^2}\]. In this form, we can more easily identify which terms to integrate. We separate variables to isolate expressions involving x on one side and y on the other. Integration techniques will allow us to find each integral. These might include:

• Direct Integrations
• Substitution Methods
• Partial Fraction Decomposition

Depending on the complexity of the terms, one or more of these techniques might be needed. For the simplified cases like in our example, direct integration is often sufficient.

Separable Differential Equations

A separable differential equation can be separated into two integrals: one involving only x and one involving only y. The given differential equation: \[(x^6 y^4 + x^2)dy = (1 - x^5 y^5 - xy)dx\] is identified to be separable. By rearranging, we isolate the y terms from the x terms, leading to:
\[\frac{dy}{dx} = \frac{1 - x^5 y^5 - xy}{x^6 y^4 + x^2}\]. By separating variables, we can write:
\[\int \frac{(terms involving y)}{(terms involving y)} dy = \int \frac{(terms involving x)}{(terms involving x)} dx\]. Each side can then be integrated independently, allowing us to solve for y explicitly. This kind of equation is often easier to handle because it breaks down the problem into manageable parts. In the end, both sides of the equation are integrated, leading to our general solution: \[\text{ln}|x| = xy + \frac{x^5 y^5}{5} + C\].