Question

the general solution to the equation. dy dx = + 4x + 1 The general solution is y(x)= ignoring lost solution

Answer 1

The general solution is y(x) = 2x² + x + C, where C represents the constant of integration, ignoring the lost solution.

Explanation:

To find the general solution to the differential equation dy/dx = 4x + 1, integrate both sides with respect to x. The integration of 4x + 1 with respect to x results in 2x²+ x + C, where C is the constant of integration. This solution represents the antiderivative of4x + 1 with an additional constant C to account for all possible solutions to the differential equation.

The differential equation dy/dx = 4x + 1 indicates the relationship between the derivative of y with respect to x and the function 4x + 1. Integrating both sides yields the family of curves represented by y(x) = 2x² + x + C, where C encompasses the constant term arising from the indefinite integral. This constant C accommodates different initial conditions or specific values of y and x to encompass various solutions.

The general solution y(x) = 2x² + x + C represents a family of functions that satisfy the given differential equation. It represents the most general form of the solution, where any particular solution can be obtained by specifying a value for the constant C. The inclusion of the constant C allows for flexibility in accommodating different initial or boundary conditions that may be present in specific problems involving this differential equation.

Answer 2

The general solution to the differential equation dy/dx = 4x + 1 is obtained by integrating both sides, resulting in y(x) = 2x^2 + x + C, where C is the constant of integration.

Step-by-step explanation:

The question asks for the general solution to the differential equation dy/dx = 4x + 1.

To find the solution, we integrate both sides with respect to x.

The integral of 4x with respect to x is 2x2, and the integral of 1 is x.

Therefore, the solution is:

y(x) = 2x2 + x + C

Here, C represents the constant of integration, which is determined by the initial conditions or additional information provided.

The general solution to the given differential equation dy/dx = 4x + 1 can be found by integrating both sides of the equation.

Integrating 4x + 1 with respect to x gives us 2x^2 + x + C, where C is the constant of integration.

So the general solution to the equation is y(x) = 2x^2 + x + C, where C is any real number.