First and Second orders od Differential Equations

First Order Differential equations

A first order differential equation is of the form:

Linear Equations:

The general general solution is given by

where

is called the integrating factor.

Separable Equations:

(1) Solve the equation g(y) = 0 which gives the constant solutions.

(2) The non-constant solutions are given by

Bernoulli Equations:

(1) Consider the new function .

(2) The new equation satisfied by v is

(3) Solve the new linear equation to find v.

(4) Back to the old function y through the substitution .

(5) If n > 1, add the solution y=0 to the ones you got in (4).

Homogenous Equations:

is homogeneous if the function f(x,y) is homogeneous, that is

By substitution, we consider the new function

The new differential equation satisfied by z is

which is a separable equation. The solutions are the constant ones f(1,z) - z =0 and the non-constant ones given by

Do not forget to go back to the old function y = xz.

Exact Equations:

is exact if

The condition of exactness insures the existence of a function F(x,y) such that

All the solutions are given by the implicit equation

Second Order Differential equations

Homogeneous Linear Equations with constant coefficients:

Write down the characteristic equation

(1) If and are distinct real numbers (this happens if ), then the general solution is

(2) If (which happens if ), then the general solution is

(3) If and are complex numbers (which happens if ), then the general solution is

where

that is

Non Homogeneous Linear Equations:

The general solution is given by

where is a particular solution and is the general solution of the associated homogeneous equation

In order to find two major techniques were developed.

Method of undetermined coefficients or Guessing Method

This method works for the equation

where a, b, and c are constant and

where is a polynomial function with degree n. In this case, we have

where

The constants and have to be determined. The power s is equal to 0 if is not a root of the characteristic equation. If is a simple root, then s=1 and s=2 if it is a double root.
Remark. If the nonhomogeneous term g(x) satisfies the following

where are of the forms cited above, then we split the original equation into N equations

then find a particular solution . A particular solution to the original equation is given by

Method of Variation of Parameters

This method works as long as we know two linearly independent solutions of the homogeneous equation

Note that this method works regardless if the coefficients are constant or not. a particular solution as

where and are functions. From this, the method got its name.
The functions and are solutions to the system:

which implies

Therefore, we have

Euler-Cauchy Equations:

where b and c are constant numbers. By substitution, set

then the new equation satisfied by y(t) is

which is a second order differential equation with constant coefficients.

(1) Write down the characteristic equation

(2) If the roots and are distinct real numbers, then the general solution is given by

(2) If the roots and are equal ( ), then the general solution is

(3) If the roots and are complex numbers, then the general solution is

where and .