Unformatted text preview: 2 ( t ) Â° = 0 . Then L [ y ] = 0 if and only if y ( t ) = c 1 y 1 ( t ) + c 2 y 2 ( t ) . Similarly, L [ y ] = f ( t ) if and only if y ( t ) = Â· c 1 y 1 ( t ) + c 2 y 2 ( t ) Â¸ + Â¹ t f ( Ï„ ) a ( Ï„ ) y 1 ( Ï„ ) y 2 ( t ) âˆ’ y 1 ( t ) y 2 ( Ï„ ) y 1 ( Ï„ ) y Â° 2 ( Ï„ ) âˆ’ y Â° 1 ( Ï„ ) y 2 ( Ï„ ) dÏ„. General Theory. Recall that we use the notation y ( k ) = d k y dt k to denote the n th derivative. We say that an equation of the form F Âº t, y, y (1) , . . . , y ( n ) Â» = 0 is an n th order diferential equation . In particular, we can express the highest order derivative in terms of the lower order derivatives: d n y dt n = G Âº t, y, y (1) , . . . , y ( n âˆ’ 1) Â» . 1...
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 Spring '09
 EdrayGoins
 Linear Equations, Equations, Derivative, Elementary algebra, dt, Linear map, c1 y1

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