Calculus Course/Differential Equations/2nd Order Differential Equations

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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