Solution of a differential equation

A solution to a differential equation is a function that satisfies the equation. In other words, if we substitute the function and its derivatives into the differential equation, the equation will be valid for all values of the independent variable in the function's domain.

For example, consider the differential equation:

`dy/dx = x`

A solution to this differential equation is a function `y(x)` that satisfies the equation. One possible solution is `y(x) = x^2/2 + C`, where C is a constant of integration. We can verify that this function is a solution by taking its derivative with respect to x and comparing it to the right-hand side of the equation:

`dy/dx = x^2/2 + C`

`d/dx (x^2/2 + C) = x`

`x = x`

Therefore, `y(x) = x^2/2 + C` is a solution to the differential equation.

There are many techniques for finding solutions to differential equations, including separating variables, integrating factors, series solutions, and numerical methods. The choice of method depends on the type and complexity of the equation, as well as the boundary conditions or initial conditions that must be satisfied.