# Calculus Course/Differential Equations/2nd Order Differential Equations

From Wikibooks, open books for an open world

## Contents

## 2nd Order Differential Equation[edit]

2nd Order Differential Equation is an equation that has the general form

## Characteristic Equation[edit]

2nd Order Differential Equations above can be rewritten as shown

Let

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle s = \frac{d}{dx}}**

Then

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle s^2 + \frac{b}{a} s + \frac{c}{a}= 0}**

- ) t

### Case 1[edit]

When

Then

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle s = e^(-\alpha t)}**

Equation has *one real* roots

### Case 2[edit]

When

Then

Equation has *two real* roots

### Case 3[edit]

When

Then

Equation has *two compex* roots

## Special Case[edit]

### Case 1[edit]

Differential Equation of the form

Roots of equation

### Case 2[edit]

Differential Equation of the form

Roots of equation

## Summary[edit]

2nd Order Differential Equation

has roots depend on the value of **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): \lambda **

With