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# 1_4 - Solutions 1.4-Page 40 Problem 7 Find general...

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Solutions 1.4-Page 40 Problem 7 Find general solutions (implicit if necessary, explicit if convenient) of the differential equations. 3 / 1 ) 64 ( xy dx dy = Separating the variables yields: dx x dy y dx x dy y 3 / 1 3 / 1 3 / 1 3 / 1 3 / 1 4 64 = = Integrating both sides to obtain yields: ) ( x y C x y C x y dx x dy y 3 2 2 3 2 3 4 3 / 4 3 / 2 3 / 4 3 / 2 3 / 1 3 / 1 + = + = = 2 / 3 3 / 4 ) ~ 2 ( ) ( C x x y + = Where C C 3 2 ~ =

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Problem 9 Find general solutions (implicit if necessary, explicit if convenient) of the differential equations. y dx dy x 2 ) 1 ( 2 = Separating the variables yields: 2 1 2 x dx y dy = Integrating both sides to obtain yields: ) ( x y 2 1 2 x dx y dy = 2 1 2 x can be decomposed into x x + + 1 1 1 1 The above integral becomes C x x C x x e e e y e y C x x y x dx x dx y dy ) 1 ln( ) 1 ( ln ) 1 ln( ) 1 ln( ) 1 ln( ) 1 ln( ln 1 1 + + + = = + + = + + = x x C x y + = 1 1 ~ ) ( Where C e C = ~
Problem 25 Find explicit particular solutions of the initial value problem. y x y dx dy x 2 2 = , 1 ) 1 ( = y The variables in the differential equation are separated as follows. ( ) dx x x dx x y dy x y dx dy x x 1 2 2 2 ) 1 2 ( ) 1 2 ( + = + = + = Integrating both sides to obtain yields: ) ( x y ( ) x e C y e y C x x y dx x y dy x C x x x 2 2 ~ ln ln 2 ln 2 1 = = + + = + = + + ~ Where C e C = The initial condition is used to solve for the constant. ( ) e C e C y 1 ~ 1 1 ~ ) 1 ( 1 = = = ) ( x y becomes e xe x y x 2 ) ( = Simplifying gives ( ) 1 2 ) ( = x xe x y

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Problem 29 (Population growth) A certain city had a population of 25000 in 1960 and a population of 30000 in 1970.
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1_4 - Solutions 1.4-Page 40 Problem 7 Find general...

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