**A first order differential equation can be solved using below mentioned methods:**

- Graphical method of plotting slope fields.
- Euler's method which provides steps to find numerical approximation to solutions of differential equations.
- Integration methods used for separable equations and linear equations.

**Solving Differential equations using slope fields:**

This is a graphical method of solving differential equations of the form $\frac{dy}{dx}$ = F(x, y). This method is based on the fact that the derivative of a function represents the slope of the tangent line at that point. Slopes at various points are calculated using the function F(x, y) and these values are represented at the corresponding points by short line segments. These line segments are called slope fields or direction fields and indicate the direction of the curve at that point. Using direction fields the general shape of a curve can be visualized and sketched.

The graph of slope fields for the function $\frac{dy}{dx}$ = x^{2} + y^{2} - 2 is shown below with the curve sketched for a given initial value.

__Euler's Method:__

Euler's Method uses the idea of direction fields and uses a series of Linearizations to find approximate numerical solutions to differential equations. Suppose y' = F(x, y) with initial value (x_{0}, y_{0}) and we need to find approximate solutions to equally spaced x values like x_{0},

x_{1} = x_{0} + h, x_{2} = x_{1} + h, -------------, x_{n} = x_{n-1} + h.

The step wise approximations are done using,

y_{1} = y_{0} + hF(x_{0}, y_{0})

y_{2} = y_{1} + hF(x_{1}, y_{1})

---------------------------

y_{n} = y_{n-1} + hF(x_{n-1}, y_{n-1})

where 'h' is the step value allowed to get the next point. The accuracy of approximation improves by reducing the value of h.

__Separable Equations:__

A separable first order differential equation is of type $\frac{dy}{dx}$ = f(x).g(y).

If g(y) ≠ 0, we can take h(y) = $\frac{1}{g(y)}$ and the equation can be equivalently written in as

$\frac{dy}{dx}$ = $\frac{f(x)}{h(y)}$

Now the functions of each variable can be grouped with the corresponding differentials on either side of the equation.

h(y) dy = f(x) dx (Variables separated)

The general solution to the differential equation can be got by integrating the variables separated equation in the form

y = p(x) + C. The value for the "C", the constant of integration and hence the the particular solution can be found using the initial value (x_{0}, y_{0}).

**Linear first order differential equations:**

A first order linear differential equation is of the form $\frac{dy}{dx}$ + P(x).y = Q(x)

where P and Q are continuous functions of x in the given interval.

This equation can be solved by multiplying the equation by the integrating factor = $e^{\int Pdx}$

$e^{\int Pdx}$ ( $\frac{dy}{dx}$ + P(x).y) = Q(x).$e^{\int Pdx}$

On integration the equation will be reduced to

$e^{\int Pdx}$.y = $\int e^{\int Pdx}Q(x)dx$

The integration on the right side can be easily performed which will lead to the general solution of the given equation.