Solution of Equations

 

Polynomic Equations

Quadratics

 

Cubics

 

Let

 

The three solutions are

 

Differential Equations

Separation of Variables

 

Exact Equation

††††

 

Linear First Order Equation

 

Bernoulliís Equation

Let

Apply the recursive formula,

If , the solution is

 

Homogeneous Equation

Let

For , the solution is

If , the solution is

 

Linear, Homogeneous, Second Order Equation

b and c are real constants

Find the roots, and , of the equation,

There are three cases:

Case 1:and are real and distinct

Case 2:and are real and equal

Case 3:and are imaginary

Let

The solution is

 

Linear, Nonhomogeneous, Second Order Equation

b and c are real constants

Find the roots, and , of the equation,

There are three cases:

Case 1:and are real and distinct

Case 2:and are real and equal

Case 3:and are imaginary

Let

The solution is

Euler or Cauchy Equation

Let

The equation becomes a linear second-order equation

 

Besselís Equation

 

Transformed Besselís Equation

Let

 

Legendreís Equation