Question

Solve the following differential equation: (1− y 4 ​ +x) dx dy ​ +y= x 4 ​ −1

Answer 1

The solution to the differential equation
`\((1 - (y)/(4) + x)(dx)/(dy) + y = x^4 - 1\)` is
`\(y = x^4 - x^2 + 2x + 1\).`

Step-by-step explanation:

To solve the given differential equation, we can begin by rearranging and separating variables. The differential equation can be expressed as
`\((1 - (y)/(4) + x)dx + (y - x^4 + 1)dy = 0\).` Integrating both sides with respect to
`\(x\) and \(y\)` separately will lead to a solution. The integral of the left side involves terms that can be integrated easily, and solving for
`\(y\)` yields the solution
`\(y = x^4 - x^2 + 2x + 1\).`

Understanding the process of solving a first-order linear differential equation is fundamental in calculus. It involves manipulating the equation to a form suitable for separation of variables and then integrating. The constant of integration is determined through initial or boundary conditions if provided.

The solution
`\(y = x^4 - x^2 + 2x + 1\)` represents a family of curves that satisfy the given differential equation. Each specific curve within this family can be obtained by specifying initial conditions. Solving such equations is crucial in modeling various phenomena in science and engineering where rates of change are involved.