Question

Find the general solution of the following differential equation. Primes denote derivatives with respect to x. x(4x+3y)y⁴ +y(12x+3y)=0 The general solution is (Type an implicit general solution in the form F(x,y)=C, where C is an arbitrary constant. Do not explicitly include arguments of functions in your answer)

Answer 1

The general solution of the following differential equation is:

`\[x^2y^5 + (1)/(2)y^6 = C\]`

Step-by-step explanation:

The given differential equation is a nonlinear first-order ordinary differential equation (ODE). To find the general solution, we can use the method of separation of variables. By rearranging terms and integrating both sides, we arrive at the implicit general solution above.

First, notice that the given ODE can be rewritten as
`\((4x + 3y)y^4dx + (12x + 3y)dy = 0\).`Now, separate variables by dividing through by
`\((4x + 3y)y^4\),` and integrate each side. The integration process involves recognizing the left side as a perfect differential, simplifying the integration steps. After integrating, we obtain the general solution in the form
`\(x^2y^5 + (1)/(2)y^6 = C\), where \(C\)` is an arbitrary constant.

This final expression represents a family of curves that satisfy the given differential equation. The constant
`\(C\)` incorporates all possible initial conditions or constraints, capturing the entire solution space. The process involves algebraic manipulations and integration steps, standard techniques in solving first-order ODEs.