Question

Find general solution y=y(x) of the followig homogenous differential quations

Answer 1

The general solution to the homogeneous differential equation
`\((dy)/(dx) - 2y = 0\)` is given by
`\(y(x) = Ce^(2x)\)`, where (C) is an arbitrary constant.

Step-by-step explanation:

The given differential equation is a first-order linear homogeneous differential equation. To find the general solution, we can rewrite it in standard form:
`\((dy)/(dx) - 2y = 0\)`. The solution to this type of differential equation is commonly expressed in the form (y(x) = Ce
`^(kx)\)`, where (C) is a constant and (k) is the coefficient of (x). In this case, (k = 2), so the general solution is
`\(y(x) = Ce^(2x)\)`.

In the solution, the constant (C) represents the arbitrary constant of integration. This constant is determined by any initial conditions provided or through further information about the problem. The exponential term
`\(e^(2x)\)` arises from the fact that the differential equation is linear with a constant coefficient. The solution
`\(y(x) = Ce^(2x)\)` represents a family of functions that satisfies the given differential equation.

In summary, the general solution to the homogeneous differential equation
`\((dy)/(dx) - 2y = 0\) is \(y(x) = Ce^(2x)\), where \(C\)`is an arbitrary constant that can be determined based on specific initial conditions or additional information about the problem.

Full Question:

Find the general solution (y=y(x)) of the following homogeneous differential equation:

`\[ (dy)/(dx) - 2y = 0 \]`