# Category: First-Order ODE

## Separable Differential Equations

Consider the differential equation y'=f(t,y), or (dy)/(dt)=f(t,y).

If the function f(t,y) can be written as the product of the function g(t) (function that depends only on t) and the function u(y) (function that depends only on y), such a differential equation is called separable.

## Homogeneous Equations

If in the differential equation y'=f(t,y), the function f(t,y) has the property that f(at,ay)=f(t,y), such a differential equation is called homogeneous.

It can be transformed into a separable equation using the substitution  y=ut along with the corresponding derivative (dy)/(dt)=(du)/(dt)t+u. The resulting equation is solved as a separable equation, and the required solution is obtained by back subtitution.

## Exact Equations

The differential equation M(x,y)dx+N(x,y)dy=0 is exact, if there exists a function f such that df=M(x,y)dx+N(x,y)dy.

In this case, the equation can be rewritten as df=0, which gives the solution f=C.

## Linear Differential Equations

A first-order linear differential equation has the form y'+p(t)y=q(t).

To solve it, rewrite it in the differential form: (dy)/(dt)+p(t)y=q(t), or (p(t)y-q(t))dt+dy=0.

Now, using the facts about exact equations, we see that this equation is not exact. However, since M(t,y)=p(t)y-q(t) and N(t,y)=1, we have that (partial M)/(partial y)=p(t) and (partial N)/(partial x)=0. So, 1/N((partial M)/(partial y)-(partial N)/(partial x))=1/1 (p(t)-0)=p(t). Thus, the integrating factor is I(t,y)=e^(int p(t) dt).

## Bernoulli Equations

A Bernoulli equation has the form y'+p(t)y=q(t)y^n where n is a real number.

Using the substituion z=y^(1-n), this equation can be transformed into a linear one.

Example 1. Solve y'-3/ty=t^4y^(1/3).