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equation - Solving ODE Linear First Order Equations dy...

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Solving ODE Linear First Order Equations Form: ( 29 ( 29 t g y t p dt dy = + Solution: ( 29 ( 29 ( 29 [ ] c dt t g t t y + = f f 1 where, ( 29 ( 29 = dt t p e t f Linear Second Order Equations Homogeneous Equations With Real Constant Coefficients Form: 0 2 2 = + + cy dt dy b dt y d a Roots of 0 2 = + + c b a l l General Solution of 0 2 2 = + + cy dt dy b dt y d a Two distinct real 2 1 , l l t t e c e c 2 1 2 1 l l + Two conjugate complex b a i ± t e c t e c t t b b a a sin cos 2 1 + One double real l ( 29 t e t c c l 2 1 + Summary of Physical Laws for a Control Volume Continuity: + + = Aout Ain V C dA n V dA n V V d dt d ) ( ) ( 0 . v v v v r r r Linear Momentum: + + = Σ Aout Ain ex dA V n V dA V n V dt d F r v v r v v v v ) ( ) ( V d V C.V r r r Angular Momentum: × + × + × = Σ Aout Ain ex dA V r n V dA V r n V r dt d M r r v v r r v v v r r ) ( ) ( V d V C.V r r r Energy Equation: + + + + + + + + = - Aout Ain dA n V u V gz dA n V u V gz V d u V gz dt d W Q ) ( ) 2 ( ) ( ) 2 ( ) 2 ( 2 2 2 v v v v & & r r r
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