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Diff Eqs 2 solutions

# Diff Eqs 2 solutions - dy 5y = x7 dx dy P x y = Q x dx 1a x...

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Differential Equations 2 – Solutions 1a. 7 5 x y dx dy x = - Rearrange so that equation resembles the general form ( 29 ( 29 x Q y x P dx dy = + 6 5 x y x dx dy = - Then, the function ( 29 x x P 5 - = and the integrating factor is = - dx x Pdx e e 5 5 ln ln 5 5 - - = = = - x e e x x Multiply d.e. through by the integrating factor x y x dx dy x = - - - 6 5 5 Remember that L.H.S. can be written as dx d ( y × integrating factor ) i.e. ( 29 x yx dx d = - 5 Integrate w.r.t. x c x xdx yx + = = - 2 2 1 5 5 7 2 1 cx x y + = where c is an arbitrary constant This is the general solution. 1b. x e y dx dy 4 2 = + Compare with the general form ( 29 ( 29 x Q y x P dx dy = + The function ( 29 2 = x P

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and the integrating factor is x dx Pdx e e e 2 2 = = Multiply the d.e. through by the integrating factor x x x e ye dx dy e 6 2 2 2 = + Remember that L.H.S. can then be written as dx d ( y × integrating factor ) i.e. ( 29 x x e y e dx d 6 2 = Integrate w.r.t. x c e dx e y e x x x + = = 6 6 1 6 2 x x ce e y 2 4 6 1 - + = where c is an arbitrary constant 1c. x y x dx dy 2 1 - + = Rearrange so that equation resembles the general form of a linear equation x x y dx dy 2 1 2 2 1 - + = x x y dx dy 2 1 2 1 2 - = - This looks like the general first order linear d.e. ( 29 ( 29 x Q y x P dx dy = + with ( 29 x x P 2 1 - = The integrating factor is x dx x Pdx e e e ln 2 1 2 1 - - = = 2 1 2 1 ln - = = - x e x Multiply d.e. through by integrating factor
2 3 2 1 2 3 2 1 2 1 2 1 2 - - - - - = - x x x y dx dy x As usual, the L.H.S. can be expressed as dx d ( y × integrating factor ) i.e. ( 29 ( 29 2 3 2 1 2 1 2 1 - - - - = x x yx dx d Integrate w.r.t. x ( 29 ( 29 c x x c x x dx x x yx + + = + + = - = - - - - - 2 1 2 1 2 1 2 1 2 3 2 1 2 1 2 2 2 1 2 1 2 1 1 cx x y + + = where c is an arbitrary constant 1d. x

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Diff Eqs 2 solutions - dy 5y = x7 dx dy P x y = Q x dx 1a x...

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