1. Separate variables. 2. Integrate both sides. 3. Add the constant of integration (+C). 4. Use initial conditions to solve for C. 5. Substitute C back into the equation.

Use the given initial condition (a point (x, y)) and substitute the values into the general solution. Then, solve the equation for C.

Add the constant of integration, $+C$, to one side of the equation. This accounts for all possible solutions before applying initial conditions.

Separate the variables: $\frac{dy}{y} = x dx$.

Exponentiate both sides: $|y| = e^{x^2/2 + C} = e^C e^{x^2/2}$. Then, $y = Ae^{x^2/2}$ where $A = \pm e^C$.

Substitute $x = 0$ and $y = 2$: $2 = Ae^{0} = A$. Thus, $A = 2$, and the particular solution is $y = 2e^{x^2/2}$.

Consider both positive and negative cases, or use the initial condition to determine the correct sign.

Substitute the value of the constant back into the general solution to obtain the particular solution.

$F(x) = y_0 + \int_a^x f(t) dt$, where $F(a) = y_0$ and $\frac{dy}{dx} = f(x)$.

A general solution includes an arbitrary constant, representing a family of solutions. A particular solution is a single solution obtained by using initial conditions to find the value of the constant.

Initial conditions provide a specific point that the solution must pass through, allowing us to solve for the constant of integration and determine a unique solution.

Domain restrictions ensure that the solution is valid and meaningful. Ignoring them can lead to incorrect or undefined results, particularly when singularities or physical constraints exist.