Solution of Equations

Polynomic Equations

Quadratics

Cubics

Let

The three solutions are

Differential Equations

Separation of Variables

Exact Equation

Linear First Order Equation

Bernoulli’s 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

Bessel’s Equation

Transformed Bessel’s Equation

Let

Legendre’s Equation